How can I increase my chances of finding the Missing Poster?

I think that a good way is the Fortune Teller machine.

The Fortune Teller costs one penny, which will either cause a random fortune to be displayed on-screen or cause the machine to spawn one of its possible drops. If blown up, it will drop several random Pickups.

You can get several Trinkets from it, but unfortunately there are 80+ Trinkets in the game, so you still need luck to find it.

As pinckerman mentions, the Fortune Teller is a good way to spawn a large number of trinkets quickly, particularly if your luck stat is high.

Another great way is by using Mom's Box, an active item with a 4-room recharge that spawns a random trinket. This is nice because it's actually a pretty good active item, so you don't have to sacrifice your progression to try and farm trinkets, and because you can use it many times during a run, especially with how often batteries drop in Afterbirth. This works especially well if you're able to break the game, giving you unlimited batteries; you're pretty much guaranteed to generate Missing Poster at some point.

it can be gained slightly quicker by using the fortune machine, but do not listen to the people who say you can get it from moms box. they are wrong, you can not obtain the missing poster from moms box so do not wase your time trying

Blank card jera to duplicate a gold chest, some gold chests will drop trinkets. I simply duplicated one to check if it drops a trinket, if so duplicate the rest by the unlimited battery method and just keep opening until you eventually get missing poster!