Oh, yeah, an inflatable pad is going to shoot right out of a hammock. Like some sort of Olympic event.
But, in a car? The pad is scooting? And, rather, it's not you sliding down or off the pad in the bag? That's a little weird. As mentioned, try lessening the inflation. Also, as the pad ages and roughs up, it might lessen what was going on.
But, pad materials span the gamut. Some are great, some are just slippery, sweaty things. There's a real art to the pad material, and not all companies get it right, nor with all the different models. Some pads just irredeemably suck.
If it's the carpet in the SUV, yeah, kitchen drawer liner will fix that - car camping so a bit more weight and bulk isn't going to sink the ship. But, might also make it super squeaky, or some other strange effect.
My lightest tent and my lightest inflatable pad don't play well together. Horribly sticky and obnoxiously squeaky between the pad and the tent floor. Sort of the opposite of what you've got. I spread some rock climbing chalk around on the tent floor, and problem solved, perhaps if only temporarily, but reapplication is easy enough.