The FAA has you last reported hours from your license
So you can request a copy of your last submitted form, and go from there. If you have a lot more hours than that and no backup, then you are out of luck to my knowledge. Its very wise for all to keep a digital copy of their logbook even if its only a scanned copy. I use safelog which works great for me, as I got my logbook stolen in a flight bag a long time ago. Since then, I still record my hours on paper and update the software logbook periodically.