Suggestions for older frames that can take wider tyres . . .

If you are happy to move to discs and want a more road based geometry have a look at the kinesis 4s disc, relatively cheap and light alu frame. Mine is running 30mm road tyres there is plenty of space still - officially it's supposed to be able to take 33mm tyres but I think you could go a bit bigger. Lots of 2nd hand and discounted ones going cheap if you dig about a bit. I bought it as a winter bike but find myself riding it 90% of the time.

Try wider rims first, the difference in comfort between a 25c on a 15c rim and a 25c on a 19c rim is massive! And you won't mostly need a new bike for that.

Anything rim braked over 28mm will need long drop brakes or cantis. There are loads of the latter about but they'll either be tourers or CX bikes which won't have a position anything like your Supersix. Depends how fast you want it to be on the road and how long you intend to ride for.
Disc brakes open up lots of options.
Spesh Diverge
Giant Defy Advanced (great carbon frame, cheap secondhand), I've got 28mm GP4000s on mine on wide rims so they're probably nearer 30mm.
Trek Domane disc takes 30s (not sure from which year though)
Giant Revolt is more gravelly than the Defy
I had a Merida Cyclo-cross for a while which worked well for most things, even did Ride London on it
Fairlight Strael goes up to 32 I think, so do some of the Mason bikes. Condor Fratello is another.

The Specialized Diverge may be worth a look, it's fairly road focussed, but if you're building from a frameset, you have different options, I've made mine a little more trail oriented by changing out the groupset to Shimano GRX and running 38mm tubeless at 35psi, very smooth. 

This may not be as sporty as you require but older Pinnacle Arkoses may fit the bill. Mine is 2014 I think and is sitting on 40 mm tyres at the moment. It's no racer but not as 'gravelly' as the newer models, with a more road oriented geometry and gearing. Are you looking for just a frameset or complete bike? The former might be harder to find. Watch out for mudguard mounts at the fork crown though, there isn't one on mine.

The Ribble CGR has been around for a while now (5+ years?) so might be another one to look into. IIRC it was one of the most popular options for commuting etc. before "gravel" took over the world. 

The Croix De Fer is a good bet.

Mine is 8 years old and takes 700x40.

Sounds like you could explore Cyclocross frames - I'm excluding gravel because gravel as a marketing thing is too recent if you are thinking about budget.

We discuss what the difference is between road, gravel and cyclocross in our club, and road bikes are allowing for a lot wider tyres, but still tend to be limited - unless you have a disc road bike, the limitation is in the brakes themselves which struggle with tyres bigger - road rim brakes haven't the capacity to cope with 32.

On older frames, you will find that CX bikes allow for wide tyres and pre-disc wheels use centre pull brakes. Bikes now sold as gravel bikes are almost universally disc. CX bikes tend to have a higher bottom bracket height. There is not a lot of difference between CX and gravel now, gravel will have more relaxed steering and a lower BB and also tend to have more fitting points for touring goodies - gravel will typically have mudguard fixing points and bike rack fixings, as well as more bottle cage mount points.

It depends on your budget and how old you are going as what have evolved as gravel bikes haven't been around that long. So if you are after a one year old model, then look for gravel bikes, older than that you are in CX territory to get the spacious frame, but you may find you don't like the idea of the alternate brakes.