What's the quickest way for promotion during and after Army BCT?

Kiss a lot of butt, well at fort Irwin that’s definitely the way to go if you want rank

When I went people for promoted for having the highest PT scores and for combatives.

You likely will not get promoted during BCT. If you do it is because promotions are automatic for grades E-2 to E-4 based on time in grade and service. However in BCT, everyone is just a "private" and in my BCT Bn we were a mix of E-1 through E-4's but no one was allowed to wear rank.

learn your job, then do your job to the best of your ability. have a good attitude, be in shape.

Promotion mostly depends on the needs of the Army. If you are E-1 right out of basic, your promotion to E-2 is automatic after six months of active service. Of the four lower enlisted ranks, I was an E-2 the shortest, maybe four months, but I spent most of my time as a PFC until I was automatically promoted to Specialist. Keep your record clean. Keep your alcohol intake low. Beer and gin are stellar career destroyers from back in the 90s. Make friends with a good group of people across the posts and bear with and obey even the worst leadership examples.

Do you have any college time that be proved, pay close attention to what you are doing in your MOS, do the job assigned to you to the best of your ability. The best way to also get ahead is to do extra training in the field that you have decided.

Why would you want to get promoted quickly? You will have less time to learn your job. The key to "fast" promotion is to always pass PT and to complete your NCOES requirements as possible. As soon as you make Specialist bug your First Sergeant to send you to WLC. Good Luck!

unless you have a bachelor's degree or are an Eagle Scout or have some other qualifier that brings you in at E2 or E3, you will be promoted by time in grade (TIG) and time in service (TIS). There is a waiver process for one or both provided on a merit basis.

Talk with your recruiter about getting promotion credit before you enlist. If they won't do it, go to a different recruiter. If they agree, ask for it in writing. If it's not on your DD form 4, or a DA form 4187, it didn't happen.
If you have enough college credit, you'll automatically qualify for E-3. I believe hot can earn E-4 if you have an associates.