Why bigger? An '06 will kill anything in North America. Buy some reloading equipment and work up some loads for heavier bullets. You will be amazed at how interesting it is to work with ballistics tables and different powders with different weight bullets. Plus the knowledge you pick up will make you sound like you really do know what you are talking about.
You didn't say if you had anything besides the '06. Do you have a varmint rifle? Look at a .22-250, .243 or maybe .25-06...Fun rounds with eye popping performance...