The good news is once you get your entry level mmc, 270 a day shouldn’t be too hard to find. You may actually get started at a higher rate than that. I can’t speak for ships, but on tugs and OSVs, ordinary seaman pay is in the high 200s/low 300s.
Most of the highest paying jobs are rotational and liveaboard, so i wouldn’t get too hung up on moving. It is helpful to live somewhat close to a large airport though.
From what I hear its not uncommon for an AB who holds a tankerman making a day rate in the high 400s/low 500s on an ATB.
It will take you some time once you get your foot in the door somewhere. Centerline, Moran, Crowley, Vane, etc…
You may have to gain some seatime elsewhere and use a couple other companies as a stepping stone to move up too. A lot of places have gotten picky with hiring entry level.