Pretty much any regional. You have other options, too.

As you mentioned that you want the airlines, a regional is a no-brainer.

Logging your time by the month isn't necessarily a bad thing; I've known others to do it, mostly ag operators.

As a private pilot, however, it's a bit unusual. Generally private pilots like to log every detail.

Never the less, if you say you flew it and your ability backs up what you say you can do, it shouldn't impede you. Your goal here is ATP (the pilot certificate, not the business).