In addition to double-sided sticky tape, there's also "carpet tape" which is sticky on both sides and wide (and has a backing paper). It comes on a roll at hardware/home improvement stores, etc.
You could also use something that's sticky on one side and adhere it somehow to a piece of cardboard, etc. (whole sheets of clear adhesive sheeting for printing "labels", Contact Paper for lining drawers, etc... even things like sheets of felt sometimes come with one side adhesive and a backing sheet).
There are also sprays and liquids that will "dry" so they're still tacky:
..."repositionable" clear sprays (hardware store, etc) ...also specifically for photos like "Photo Mount"
...a white glue that dries clear but sticky... Aleene's Tack it Over and Over, or just "sizing" glue
...most any permanent white glue (like Elmer's Glue All)... those can be painted onto a stiff background, and when dry will still probably be sticky enough to press things to a bit
You could also use some of that grippy, non-slip, rubbery drawer liner on a roll since it's supposed to keep things from moving around... like this:
http://housewares.hardwarestore.com/37-188-shelf-and-drawer-liners.aspx
For some of those possibilities, you might have to seal the finished puzzle from the *front* first after you're finished with it, if it can't easily be removed from it's sticky backing, and if you want to do that.
After the front dries, remove any of the temporary sticky material and seal the back too.
HTH,
Diane B.