No plane needed. Just run a 19.5" wire down your back. That will make it a true dipole. I put these "rat tails" on all my HTs and the difference it makes is impressinve. I get several S-units more on Rx. I assume Tx is equally improved.
Actually, if your coax is about 19" long you'll have your dipole.
That is pretty much what I was going to say as a serious response, but he beat me to it.
The other option is a counter poise aka "tiger tail" which can greatly help with your signals. No true grounding necessary (unless you expect to be out in the wide open plains during a lightning storm a lot, even then grounding like you mention will not really help much anyways if you're hit).