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I wouldn't go smaller than a 3/4 inch thick piece. If you are using a floor jack you could use two pieces. One for the rear wheels and the other for the front. That would make it easier to pack. Probably would want something about 8 inches by 8 inches.
I carry a few squares of plywood anyway for jack stands on paved paddock so they don't dig into hot asphalt. Part of my travel kit. I'd cut a few and throw them in a box.
Just as a safety precaution, plywood or 2x4's under the jack should do the job for any surface other than concrete. The flat plywood spreads the load on soft surfaces like dirt or asphalt while the 2x4 seems to work best on gravel. Same procedure for jack stands too.
offroad and rally cars both utilize a "jack skidplate"google images will turn up gobs of different designs as well as premade ones for sale but heres one to get you started:
St. Jude Donor '03-'04-'05-'06-'07-'08-'09-'10-'11-'12-'13
Probably are not going to have to have room to bring it with you, but soemtimes we got stuck in the grass at Mid-Ohio, Lime Rock, or Road America we brought 6 to 8 sheets of 3/4" plywood to put under the cars. We welded full plates on the bottom of each jack stand so they had a full 'skid' plate under them so they wouldn't punch through the wood. The jack worked ok but you still need to be really careful.
Still remember changing the clutch in the C6 challenge car in the grass at Mid-Ohio....nightmare.