Had an interesting discussion with my GM last night at game.
When it comes to multiple group combats, is there any way for a combatant to gain "surprise" (basically cause their enemies to be flat-footed to their attacks) after initiative has been rolled between the two initial groups without the use of stealth, if all combat participants have already acted?
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After the initial surprise round is over, you're just adding new combatants.
Stealth, via invisibility, is the most likely way to introduce them unawares.
If your party is bad at Perception, you could have more antagonists arrive at extreme range using composite longbows (range increment of 110ft effectively means they have +11 to Stealth) and sniping.
There's a mid-level Pathfinder Module where an archer uses True Strike and sniping in that manner.
Beyond that, there are multiple ways to make your PCs flat-footed during combat, so you could just have the new enemies run around the corner and start feinting, or something.
You can't normally become Flat-Footed after combat has started.
A character who has not yet acted during a combat is flat-footed, unable to react normally to the situation. A flat-footed character loses his Dexterity bonus to AC and Combat Maneuver Defense (CMD) (if any) and cannot make attacks of opportunity, unless he has the Combat Reflexes feat or Uncanny Dodge class ability.
Characters with Uncanny Dodge retain their Dexterity bonus to their AC and can make attacks of opportunity before they have acted in the first round of combat.
In the case of a third group entering a fight, I would roll Perception vs Stealth to see which of the combatants notices the third group. Those who fail their Perception check would lose their Dex-to-AC against the first attack made by the third group's members as per the normal rules for Stealth. The combatants wouldn't be Flat-Footed since they already are in combat.
Losing Dex-to-AC represents a character not being able to react to a blow.
Being Flat-Footed is a condition which represents a character not being able to react to combat.
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Feinting and Stealth/Invisibility does not make enemies Flat-Footed, but it will make the enemies lose their Dex-to-AC.