Is there any RAW method that increases the rate of fire for a halfling sling staff?
Per the FAQ, the halfling warslinger racial trait does not because it only applies to "a sling" and not all sling types. In similar vein, the halfling feats would not apply to the halfling sling staff either, but at least apply to the double sling.
Halfling, Warslinger: What kind of slings does the this reload ability work with?
The warslinger ability says, "Halflings with this racial trait can reload a sling as a free action." It doesn't say "any type of sling" or "all slings," just "a sling." The ability only affects standard slings, not halfling sling staffs or any other kind of sling.
Reference the Halflings of Golarion feats:
Ammo Drop (Combat)
Your coordination is so perfect that you can simply drop ammunition from your hand at the exact moment required for it to fall into an open sling as your twirl it around.
Prerequisites: Sleight of Hand 1 rank, proficient with sling.
Benefit: You can load a sling or one end of a double sling with one hand as a swift action or move action. This does not provoke an attack of opportunity.
Juggle Load (Combat)
Your fingers are so nimble that reloading your sling is almost effortless for you.
Prerequisites: Ammo Drop, Sleight of Hand 1 rank, proficient with sling.
Benefit: You can load a sling or double sling as a free action. This does not provoke attacks of opportunity. This feat allows you to fire your sling as many times in a full-attack action as you could attack if you were using a bow.