What level of SQF are you attempting to attain? Anything Level 2 or 3 requires a desktop audit a minimum of 30 days in advance. So if you can have your documents done by June and get them scheduled with an SQF body, then yes, it's possible.
When we did our SQF certification, we completely converted our old SOP book to a FS-QMS system for the desktop audit. So with that in place we only had about a month of new documents to show. We were happy to show them the old programs, and we were honest and up front that SQF was a huge step forward for us, and we've improve these programs in these ways. The auditors were impressed with how far we had come, re-iterated that when the come back next year they'll expect to see a years worth of follow through on the new documents. Because the audit is a "snap shot" at the time of the audit, I think you could easily do the same. If your old documents don't meet standard, but you can show the new ones do, it sounds like you'd be in the same place we were.