Improved Critical no longer doubles your weapons' critical threat range. Instead it applies a static bonus of +1 to +3 to your range, depending on weapon type. So it doesn't matter if it applies before or after competence bonuses to critical threat range like Blade Specialization because they're just added together.
But there are two Competence bonuses to critical threat range which are doubled if you have the corresponding Improved Critical feat to go with it: Swashbuckling (any compatible weapon) and Assassin's Knife Specialization if using daggers. E.g., if you made a Swashbuckler using Forester's Brush Hook, you would have:
18-20/x3 base
17-20/x3 Swords to Plowshares feat
16-20/x3 Improved Critical (slashing)
12-20/x4 Swashbuckling (normally +2 range becomes +4 range w/ICS)
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