@The Nth Doctor I did serve in the 1980s and the Navy was offering all kinds of incentives for enlisting. Reagan was buiding the 600 ship Navy, after all.
I exited boot camp as an E-3 and A-school as an E-4 for signing up for a 4 active/2 inactive reserve Advanced Electronics Program (the most common hitch at the time was two years active/four inactive reserve.). That got rolled over to Electrician's Mate when my eyesight disqualled me for sub duty. Was an E-5 within three years. I extended so those two reserve years became active and I was working on E-6 when I separated.
But E-5 in four years without even really trying.
As for Jonesy, he most likely would have gone to sub school and then A-school, so, even without enlistment incentives, he could have been a E-3 before he even hit the boat. Then could have re-upped and asked to stay on the boat. Seaman Beaumont OTOH comes across as a striker.
The movie is not the best but it is better most every other one out there.