Looking to put some wider tires on my road bike for the winter. The fork has plenty of clearance for 28c, but unfortunately my old Shimano 600 calipers don't. Any recommendations for a caliper brake that'll offer a bit more tire clearance?

Any brakes that are labeled "long reach" should do. Shimano makes a pair called BR-R650.

my 28c tires won't fit without taking some air out. But I like the ride so much more that when ever I need to take them off, I let some air out and re-inflate once mounted. So far not too much of a hassle.

...No they won't

Long reach are just that - longer caliper arms for use with frames that have the brake bridge mounted higher than normal or for when you put a 700c wheel on a frame designed for 27". They will allow you to lower the brake pads.

The issue with caliper clearance is at the top of the caliper - or the throat of it - at the brake bridge of the frame. OP - that clearance is mostly determined by the frame itself. Running larger tires on modern frames has become an issue. Funny because now it's the fashion. OP - depends on the tire. You are going to have much more luck if you try out clearance with different tires. Just because a tire says something on it's sidewall doesn't really mean anything with regards to it's actual size in reality. don't believe it - check with calipers.

All newish brakes can probably do it. It's usually the pads and holders that reduce the clearance. Look into low profile brake shoes.

Sram Apex brakes are supposed to clear 28mm tires, if your frame allows it.

Well, I tried out a pair of sram apex calipers with some 28mm gatorskins, and the top of the front caliper is still skirting the tire (although the apex are a significant upgrade). There's plenty of daylight between the fork crown and the tire, but the angle at which the brake protrudes kills the clearance. Oh well, unless someone has another idea, I guess I'm back to the 25mm up front.

Loads of room in rear, btw.

I have 28mm Schalbe Marathon tires on my 1999 Litespeed Vortex. The brakes are 1999 Campy Record. I have put the same wheels/tires on my Colnago EPS with 2006 Campy Record brakes. The wheels have Hed rims, which are wider and maybe cause the tire to be less tall. You might also try a different brand/model of tire. Not all 28mm tires have the same dimensions.

The Cane Creek SCR-3 brakes I have on one of my bikes clear 28mm Continental Ultrasport and 28mm Vittoria Randonneur Cross tires with room to spare.