I got out of the Army with an RE code of 4, for a locally imposed bar to reenlistment, in 91. (I had been a problem child and rightfully deserved every bit of punishment I received, including my bar.) Within a couple years, I was trying to get back into the Army. Unfortunately, the Army was going through the reduction in forces after the Cold War ended and wasn't taking very many prior service personnel, especially not problem children. It didn't help that the recruiter who was supposed to be helping me didn't even fill out all of the paperwork necessary to request a waiver-he just left it in his desk drawer.
Fast forward to the summer of 96 and another recruiter took over the effort to help me reenlist. Because of my bar, I had to get the recruiting company commander to sign off on a waiver. I had to meet with him, answer a bunch of questions (a lot about why I got the bar, my alcohol use since leaving the Army, my plans if I were able to reenlist, and why I thought he should give me a second chance), and provide him with several letters of reference from members of the community in good standing (I used a business owner I knew, a former teacher of mine, and the local American Legion Commander(a retired Navy LtCdr and retired deputy sheriff).). He pointed out to me that it was his name and reputation going on the line and that if he signed my waiver, I better not let him down. I assured him I wouldn't and he signed it.
The waiver gave me 90 days to reenlist, or I'd have to go through the process all over again. Luckily for me, he signed it just before the end of the current fiscal year / start of the new one. Once the new fiscal year started, USAREC opened a VERY limited number of slots for prior service personnel, all of which were expected to go VERY quickly (within days, if not hours). I went to MEPS immediately, chose an MOS, and swore my oath, set to leave for Basic towards the end of November. When I went to MEPS, I wanted 11B, my former MOS. But, since I was going in during the time between high schools graduating and Christmas Exodus, there weren't any slots available. And, I didn't have enough time left on my waiver to wait until after New Year's-I had to be in the Army before Christmas, or I'd have to wait until the following year, at least. I ended up choosing 13F, as it was the closest I could get to 11B.
Another thing, like I said in the last paragraph, I had to report to Basic-even though I'd obviously graduated 11B OSUT. At the time, if you had a break in service longer than 3 years, excluding Guard/Reserve time, you had to go back through Basic. I guess they figured you needed to relearn how to march, strip an M-16, and, more importantly, you probably needed to get back in shape after sitting on your couch. I know I needed that part of it, lol. Of course, going through Basic a second time was a breeze, especially when the Drill Sgt's made me the platoon guide.
So, yes, it can be done, though it isn't easy. Since the fiscal year ends soon, though, you may not have time to get it done this year. But, don't let that dissuade you. Get started on it by talking to the recruiters and finding out what would need to happen and if it's even remotely possible. You might even find the Army opening slots for prior service during the fiscal year, depending on how recruiting efforts are going.