Some hospital don't even do pre registration. My hospital says to just go in when you're in labor. So I think giving the hospital a call would be a good idea, they'll let you know when and if you can.
Im pregnant with my 3rd and have never registered with my hospital or was even told I had to. Thats in CA and AZ so not sure about every state.
My hospital has online pre-registration. It will cut down on paperwork when you get to the hospital. See if your hospital has a website. I am 32 weeks but I registered two weeks ago.
My hospital says 30 days before your due date
My hospital actually called me today to pre register. I'm a ftm and didn't even know you do that lol. It was a very nice surprise. I'm due next week :-)
My midwife reminded me to do it as my "homework" between my 27 and 31 week visits. I did it much later last time around, week 35ish, but it makes sense to get it over with - in case something happens, and so you don't forget and have to do all that paperwork while in labor.