What a truly fantastic upgrade to the original Colt Vest Pocket! The level zero Colt Vest Pocket was always held back first and foremost by its need to be put into play on the same turn you intended to use it, which relegated its use to being put into play by cards like Joey "The Rat" Vigil, Sleight of Hand, or Fence. If you spent an action to play it, you would typically only have two left to fight with it, not to mention the fact that doing so could trigger an attack of opportunity. For the level two version, you can play the Colt Vest Pocket the same way you do most weapons: on any turn where you have an action to spare to prepare you for future enemy cards. The +2 combat is welcome too for a faction filled with 3 combat investigators who likely want to spend at least some time fighting enemies. However, I remain unconvinced that this gun - even in its upgraded form here - has a place at most Arkham Horror tables.
There is still some difficulty in taking full advantage of the Colt Vest Pocket. You still only have one round in which to use its 5 ammo. Rogues are obviously the class for extending a turn with additional actions (Leo De Luca, Quick Thinking, Haste...) so it is plausible you could play this and spend four or even five actions fighting with it on a future turn, but then there just aren't that many enemies with that much health! One thing you can do if you are playing Leo Anderson, "Skids" O'Toole, Tony Morgan, or Jenny Barnes is play "Eat Lead!" to shore up one of your attacks, and of course the ammunition from the Colt Vest Pocket is much more expendable than that from other guns. "Skids" and Tony get extra actions to shoot the new Colt, and Jenny's Twin .45s can also end up with plenty of ammo, so I think for these three investigators who can also take "Eat Lead!", this is a convincing enough combo to be worth considering. As for other cards that grant additional actions, I think Haste is going to be the most effective card at extending the gun's reach because the Colt highly incentivises taking multiple actions of the same type in a single turn, so they're a natural pair.
So: the level two Colt Vest Pocket is most at home in an investigator with actions to spare who can spend its ammo either taking on something with a large health pool, or for fighting lots of enemies at once in a round where another gun might run dry. The former is going to be campaign or scenario dependent, but you can make the latter more likely by playing with more players. The problem is that this role is traditionally filled by higher level weapons that do more damage per shot than the Colt Vest Pocket, such as the Chicago Typewriter or the new Beretta M1918. This is typically considered better than spending more actions fighting for less damage because you take fewer skill tests, each of which comes with a chance for things to go wrong with a failure. The Colt is half the experience cost of each of those big guns, but then if you are supporting it with other cards with levels highter than zero, such as Haste, you could have just purchased a bigger gun and had fewer pieces to assemble! Now, this isn't necessarily an either/or, as many particularly fighty investigators, such as Tony Morgan, are going to want plenty of weapons, but it bears thinking about how you might spend your experience most efficiently.
I think for now this is going to be a pretty niche pick, but this is certainly a much more broadly playable card than its level 0 variant. Perhaps in future there will be a scenario where a large enough party of investigators are guaranteed to draw multiple two health enemies in a single round, and the Colt Vest Pocket (and the .45 Thompson) will be likely to outperform the higher level Rogue guns. Until then, this is a reasonable if a bit wonky sidearm you might take if you want a particularly high density of weapons.