Ways to increase threat range
The Keen (slashing/piercing) or Impact (bludgeoning) weapon prefixes double the weapon's base threat range. A Kukri (base threat range 18-20) with Keen, for example, will have a threat range of 15-20. This does not stack with Improved Critical.
The Improved Critical feat doubles the base threat range for any equipped weapon in a specific category (slashing/piercing/bludgeoning/ranged/thrown). This does not stack with Keen/Impact.
Competence bonus to threat range can come from various sources. It affects weapon's base multiplier and thus the effect is doubled by Improved Critical feat or Keen enchantment. Multiple effects granting competence bonus don't stack.
Fighter's Keen Edge enhancement adds 1 to the threat range of a his/her selected weapon group.
Bard's Swashbuckling enhancement modifies the threat / multiplier as follows
Dagger, Shortsword, Throwing Dagger: +1 Threat, +1 Multiplier
Light Mace, Light Hammer, Kama, Sickle, Dart, Shuriken, Throwing Axe, Throwing Hammer: +2 Threat, +1 Multiplier
Kukri, Rapier: +1 Multiplier
Handaxe: +2 Threat
Light Pick: +1 Threat
Staff Specialization, available for Rogues and Monks, add +1 to thread and multiplier for quarterstaves
Monk's Ninja Master enhancement modifies the threat as follows
Shortsword: +1 threat
Kama, Shuriken: +2 threat
Legendary Dreadnought's Pulverizer adds +1 threat for bludgeoning weapons. Find details and exceptions in the LD article.
Sacred bonus (?) to threat range is available to Divine Crusaders: +1 threat for all weapons (not handwraps/animal forms). The effect is doubled by Improved Critical feat or Keen enchantment.
Insight bonus to threat range is available from Bard's Exploit Weakness enhancement. Every time you don't crit, you get a +1. The effect is doubled by Improved Critical feat or Keen enchantment.
Barbarian's Critical Rage gives a stacking +2 threat while raging. This bonus is NOT doubled with Improved Critical/Keen. (Inconsistent, might be different for thrown weapons post)