would a .30-06 give me a good bit more distance/power than the .270?
No. Not in your situation.
Your .270 is fine for any big game hunting in North America regardless if you are in dense woods or not.
A 30-30 is often called a 'brush-gun' because it isn't suited for long shots in open country. Your .270 will vastly outperform it from 0 -yards to as far as you can possibly shoot it.