Or is re-enlistment only for enlisted personnel? If officers don't re-enlist, what IS the word for when they sign up for more?
An officer's commission is not for a set block of time, the way an enlisted person's enlistment is. It's indefinite, though there is usually a minimum amount of time they must serve, depending on how they receive their commission, mostly having to do with how much the military had to invest in time and money to train them. The same holds true for enlisted people. For example, the job I did in the service (I was a Russian interpreter) had a really long training. A little over a year and a half. You can initially enlist in the USAF for just two years, but if you want to be a Russian 'terp, your minimum to get that job is an initial 4 year enlistment. The 2 year enlistment is not open to you if you want that job. When an officer receives a promotion, they recite the Services Oath, and this would be the closest equivalent to what an enlisted person does when they reenlist and swear to serve, blah, blah, blah.
Oh, darn. That doesn't work for my story! But, still - good to catch it now rather than later - thanks!
Also, whereas an enlisted person would "re-enlist" for say, 4 more years; an officer would, as an example, accept a promotion to the rank of Major for an OBLISERV (Obligated Service) of 4 years. As well, enlisted personnel are only "advanced in paygrade", whereas officers are "promoted in rank". Once an officer has completed their OBLISERV they can resign their commision at any time. So as an example one might say: "Lieutenant Smith fulfilled his OBLISERV as the Gunnery Officer aboard the USS Barnacle, but since he had no solid plans upon returning home to Muncee, Indiana; he decided to accept a 2 year billet as Chief Engineer on the USS Kelp for an OBLISERV of 4 more years." Hope this helps a little.
If it's of any help, over on this side of the pond, officers re-engage or extend their commission i.e. they accept a further number of years service, often on promotion. Same with what you may refer to as an enlisted man, over here one would 'sign on' for a further three years after the initial six, for example. In either case, it's not leaving the service and re-enlisting - it's effectively re-engaging for further service. It's the terminology used.