
Hendelbolaf |

Obviously there is Improved Critical (Unarmed Strike) and the other Critical Focus could apply as well. I cannot recall if there is anything that just improves the x multiplier, but there might be among the various styles and moves.

sk8r_dan_man |

Improved Critical (Unarmed Strike) will increase threat range to 19-20
Keen Amulet of Mighty Fists or other similar magical item will also increase threat range the same way, but the two effects don't stack.
Critical Focus will help you to confirm critical hits, and there are loads of follow up feats that will add various effects on a critical hit, but I don't think there is any that increases the damage multiplier.
After you've doubled your threat range to 19-20, the only thing you can really do to improve crit rate is to roll more d20s. More rolls = more chances to crit.
Perfect Strike effectively increases your unarmed threat range because when you roll 2d20, the chances of one of them being 20 is 9.75%. If you have 19-20 threat range, then it becomes 19%. Of course, this ability can only be used once per round, and a few times per day.
If you're playing a Weapon Adept Monk, or Zen Archer Monk, then you get Perfect Strike as a class ability that improves to let you roll 3d20 at 10th level. This results in a crit chance of 14.2625% at 20 threat range, or 27.1% at 19-20 threat range.
Haste or Monk's Ki Strike will give you one additional attack at highest Attack Bonus at the start of a flurry, which helps; they don't stack. Of course, Flurry of blows will also increase your chances of threatening a critical simply because you're rolling so many attacks. A lvl 20 Weapon Adept Monk with 19-20 threat range using perfect strike and Ki Strike with his flurry of Blows rolls 10d20, and there's about 65.1% chance that at least one of those attacks will threaten a critical. Of course, the last 3 are probably unlikely to confirm, even with Critical Focus, but at least they still hit.
That's pretty much all you can do to increase crit rate.
Improved Critical (Unarmed Strike)
Critical Focus
Perfect Strike
Full attack actions

Skull |

There is not much to increase unarmed strike crit range.
Keen only works on piercing or slashing weapons. So you will need either boar or tiger style to get those damage types. As stated it does not stack with the imp crit feat.
The extra attack from hadte has nothing to do with the extra attack you can get by spending 1 ki. You can get both.
For more extra attacks: look at taking medusa's wrath as a bonus feat later on. Combined with a staggering crit. You can get two extra attacks in after a crit. Or on any of the conditions that allow those attacks.

Dabbler |

Is there a feat which improves the base threat range or multiplier of unarmed strikes?
Just curious since if you make an unarmed fighter, you end up ceding the whole critical tree since a 20/x2 isn't really worth modifying.
No. I worked on a re-write of the monk here that used increased threat range and static bonus instead of increasing dice-type, but that's in no way official.
However, if you want more critical hits as a monk, then you need Improved Critical and the Snake Style feats. Why Snake Style? Because of Snake Sidewind, which allows you to make a Sense Motive check to confirm threats rather than a second attack roll. That means those fluke 20's you roll with your lowest attack bonus have a very real chance of confirming...
Consider:
Attack bonus (at best) = Level-2 + {Dex or Str Mod} - {iterative penalty} + {enhancement bonus up to +5}
Sense Motive (at best) = {Level skill ranks +3} + {Wis bonus} + {other bonuses, up to +6 from Skill Focus}
It's relatively easy for a monk to have a better Sense Motive check than they have an attack roll, and always better than their iterative attacks, making this one feat better than Critical Focus by far (in fact I have a sneaking suspicion that Critical Focus can stack with the Sense Motive check).

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There really isn't much to improve it other than better confirmations with Snake Style and a lot of Attack Bonus to really guarantee that Critical Confirmation. Like Dabbler said.
For Crit Range it's really going to just stop at 19-20 with Improved Critical, Keen(AoMF), etc.
Dabbler, I also am fairly positive Critical Focus would stack with Snake Style suite.

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Perfect Strike effectively increases your unarmed threat range because when you roll 2d20, the chances of one of them being 20 is 9.75%. If you have 19-20 threat range, then it becomes 19%. Of course, this ability can only be used once per round, and a few times per day.
RAW, perfect Strike cannot be used with UAS (unless they've repealed the previous FAQ).

Hendelbolaf |

Keen Amulet of Mighty Fists or other similar magical item will also increase threat range the same way
This would only work if you had a way of converting your Unarmed Strike to piercing or slashing damage, such as Snake Style. etc.

The King Of All Night's Dreaming |
Obviously there is Improved Critical (Unarmed Strike) and the other Critical Focus could apply as well. I cannot recall if there is anything that just improves the x multiplier, but there might be among the various styles and moves.
For a monk, the best way to improve unarmed attack crits would be to use the Rokugan Campaign setting book, plus The Way Of The Open Hand splashbook for the Rokugan Campaign Setting.
You need to be Phoenix Clan, then you can take the following feats representing your clan's monastic order fighting style, in order (Kiho feats are basically like 'enlightened' feats. If you are using the Rokugan flavour of monk, the Inkyo monk, that guy gets 1 [Kiho] bonus feat at 1st level, and then 1 more at each even-numbered level, such as 2, 4, 6, 8... etc. If you aren't using a Enlightened Monk, just take them normally):
Kinenhi Novice [Phoenix Kiho Feat]
Prerequisites: Int 13+, Improved Unarmed Strike
Benefit: You gain a +2 shield bonus to AC while fighting unarmed.
Kinenhi Intermediate [Phoenix Kiho Feat] (Way of the Open Hand)
Prerequisites: Base attack bonus +2, Int 13+, Improved Unarmed Strike, Kinenhi Novice
Benefit: You gain a +6 bonus to confirm critical hits with unarmed strikes.
Kinenhi Advanced [Phoenix Kiho Feat] (Way of the Open Hand)
Prerequisites: Base attack bonus +4, Int 13+, Improved Unarmed Strike, Kinenhi Intermediate
Benefit: You gain a +4 shield bonus to AC while fighting unarmed. At the beginning of your turn, you may forego this bonus to increase the critical threat range of your unarmed strikes by +1 until the beginning of your next turn. This bonus is applied before that of the Improved Critical feat.
Now, the wording on that feat makes me wonder... I get that it raises your critical threat range from 20/*2 to 19-20/*2 but what does it means that it is applied before the bonus from the Improved Critical feat? Does Improved Critical look at it and goes "Yeah, 17-20/*2 seems legit to me"? I would NORMALLY say it just goes to 18-20/*2 but that "before improved critical" thing bugs the hell outta me.
By the way, the next feat in the chain is the Kinenji Expert which goes something like "whenever your attack threatens to be a critical hit, you may choose not to confirm it, deal damage normally and then get a free unarmed attack -not neccessarily against the same target- which provokes AoO's from enemies around you".
And then comes Kinenhi Master, which goes something like "At the beginning of your turn, you may take a penalty to AC and unarmed attack rolls up to your bab. This penalty lasts until the beginning of your next turn. If you threaten a critical hit with an unarmed strike during this turn, you can forego the extra damage and instead force a will save out of the opponent with DC equal to 10 + (double your wis mod) + a bonus equal to the penalty you took. If they fail, they're stunned until the beginning of your next turn.
And of course creatures immune to criticals are immune to this effect.
That's the most far-fetched way I can think of to boost criticals with unarmed strikes.
Then there's improved critical and the other more sane options like improved critical. Honestly I think going down the chain to just expert is fine.