If for some reason your driving range provides only mats, do not hit golf balls off the mat. Hit them off a tee.
In my experience, I hit the ball much better–straighter and more accurately–off grass than off an artificial surface.
The reason, I am guessing, is that the price of a mishit on a mat is the shock of hitting it when I hit fat even a bit (and we all do that, don’t we). I don’t like that. It doesn’t feel good. So I unconsciously end up swinging not to develop my ball-striking skills, but to avoid whacking the hard mat.
The only price I pay for hitting a bit fat on grass is being somewhat chagrined.
The rubber tees at driving ranges are all tall ones designed for hitting your 460cc driver. Get a short one, maybe 1½ inches, and bring it with you. It will poke its little head through the mat just enough to lift the ball off the ground but not so much that you’ll hit the ball too high on the clubface.
The most difficult challenge this game presents to us is the ground. If you take the ground out of play, you can concentrate on what you’re there for–improving your swing–and not to avoid uncomfortable contact with a hard mat.
Speaking to that point, I would suggest using a tee even if you do hit off grass at the range.
But maybe all this is just me.
Please don’t avoid trying this idea because you think it’s cheating, though. Nick Price wrote a book on how he developed his golf swing. In it he said he hit a lot of 7-irons when he did it. Off a tee. That encourages you to swing through the ball and not worry about picking it off the ground.