I will once I find a tire that's wide enough. So far its either undersized or overpriced.
It don't make sense to spend $30 to $60 on a pair of tires online if they're not going to be any wider than the many used 27x1 1/4" tires I've got here. So far all of the 27x1 3/8" tires I've found don't even measure 1 1/4" wide, most don't measure 1 1/8" wide.
I find the same thing with 26x1 3/8" tires. I've got a Raleigh Sports here with original tires from 1967, still in good shape, I've yet to find a modern tire that's as wide as the originals, the new tires don't even look right, let alone ride right.
I've got a set of original Raleigh Record tires on 1967 Raleigh Sprite that are on par with some middleweight tires when it comes to width.

Keep in mind that the rims I'm trying to fit these to are extra wide, almost double what most 27" rims are. They're 1.5x the width of say a Schwinn S6, and nearly double the width of the more common (steel), Araya rims.
The bike is all original, and its basically going to stay that way, so the steel rims are staying. I may convert it to a single speed but I haven't decided on that yet. I did consider swapping the rear hub out for a three speed internal gear hub but the bike is heavy enough with everything else being steel.