I'm Torn

In general, finding a good 360 for a low price is a ton easier than a decent 340 to rebuild. But the eyes of the general public will open wider when you say those magic 3 numbers: '340'. (It sure does when my son mentions the 340 in his '65 'Cuda.) That may have a high personal value to you.

The 360/5.9L has more piston choices than any of them and gives you better low end torque out of the box.

A lot of this has to do with what you mean by 'more'. OP, can you help us with what you intend to do with this engine? Cruise? Drag race? Road race? 'More' than the stock 318 is at least going to be easy to accomplish.

BTW, as for budget, my son and I spent around $5k on his 340 with getting the block cheap ($300; that is good for a 340 block), using good breathing Edlebrock heads, having the machine work done locally, using a used crank, buying new chain, rockers, pushrods, pistons, rods, intake, cam, oil pump, and carb, and doing all the detailed measurements and assembly work ourselves. We had a new water pump, pan, timing cover, headers and distributor already. So you can see how the comment of the cost 'spirals out of control' can be true.