Just bought a 1963 544 Sport, original owner car.
Have no hub caps or trim rings (would like to buy some maybe)

When using aftermarket 5x4.5 bolt pattern 6"x15" rims (chrome smoothies with 3.75" backspace) what tire size will not rub ?

I have a '64.

A couple of posters have said that larger wheels and tires have made their cars handle poorer.

Try eBay - I got nice hubcaps, trim rings and a set of station wagon wheels - somewhat heavier and I think a bit wider than stock.

A few years ago Road & Track tested a BMW series 3 with +1, +2, and +3 wheels and tires (16, 17, and 18 inch)

The results - to their surprise, the original 15 inch wheels performed somewhat better than the larger sizes.

They talked to an engineer at BMW about their findings - He told them - Of course, you must realize that the car was designed for 15 inch wheels.

Their conclusion - owners attribute better performance to larger wheels, but actually the improvement comes from newer, high performance tires!

FYI a Ford Ranger pickup rim is a nice upgrade, steel or aluminum, cheap at the local salvage yard they bolt right on

For my money, I would buy trim rings and hub caps for the stock rims and put
165 x 15 tires on them. No problems, car will handle good and look like a 544 as it should. A good investment would be a larger front sway bar from IPD. Don't buy the rear sway bar as it doesn't help handling.
Another great upgrade is an OD tranny. Tom 65 544 Sport