The 270 is already overbore. What would you consider to be an improvement? Re-barreling to 280 would be an improvement because of the 9:1 twist. That reduces the effects of overbore by allowing heavier bullets which can improve long-range trajectory; however, I don't know if that is what you're looking for.
I have found the 270 to be an excellent long range cartridge.
I practice at the range with rifles in 6mmBR, .243Win, 257 Roberts AI, 260Rem, 6.5x55, 270, 7x57mm, 7mm Rem mag, 30-30, 303Sav, .308, 30-06, 300 Win Mag, 7.5 Swiss, 7.62x39mm, 303Brit, 7.62x54R, 8x57mm, 338WM, 10mm, 10.4mm, 44 magRifle , 45acpRifle 45Colt, and 45/70.
But the 270 is the only one that kills the deer for me.
Go to the local Hobby Shop and buy a piece of brass tube. Take a 270 MT cartridge to select the size of tube which will will fit over the case neck.
This will be your "drop tube". The load the 270 Win, starting without using the drop tube getting as much of IMR 7828ssc as you think is a good starting load for the 140 grain bullet you will use.
Keep increasing powder, using the drop tube, until you get decent pressure and check the velocity. You may be surprised, in a pleasant way, with the potential of that powder in the winny. AGAIN, be careful........
If the bbl is shot out, then a 270 WSM or 7mm WSM wouuld be nifty.
No doubt the 270 win is hard to improve on, but it is possible. Roy... not sure what in the heck you're talking about. Not even a guess! This "threads come off with the barrel" thing Win69 is talking about has me for a loop too. Are the threads in the action stripped when trying to remove the barrel? Love those WSM's too, but will those cartridges fit in that action?
As a game rifle, it's VERY hard to improve on the .270. (But, IMHO, the .30-06 would be as step up! Call it a .30-270! )
The AI version is pretty much pointless, it doesn't do anything useful on light and medium (for caliber) bullets and few hunters use the heavy bullets that can take a little be of advantage from the larger capacity.
Converting a .270 action to a magnum would be expensive. Re-cutting the bolt face and shaving the feed rails would be required and can (usually) be done, at a price. And the magazine box would likely be a bad fit too.
The best cartridge for your action and purpose would be the 7mm JRS. It will virually duplicate the 7mm rem mag based off a 280 rem case. 280 AI also good. To answer your question about your action, the Browning A-bolt is probably the worst action made and most gunsmiths will not fool with them. I will not. Bottom line it is a P-poor action at best. I would sell it and get another action to work with.