Best Tires for the Triumph Spitfire 1500

Got these on an older classic car. Previous owner mentioned they were ok, but I'd probably want to change them. Boy, was he right. They're tires in the academic sense - they're black, round, and made of rubber. You can put air in them and drive the car. But the sidewall flex on these things is absurd. Going around normal intersection turns, the back end felt like it was squishing around on water balloons to the point that passengers with no deep car knowledge commented on it. The tires being still relatively new, I thought the rear suspension was in need of a refresh, but when I started getting into it, I discovered it had recently been already done. So I gave the car a hip bump on the side, and was surprised at how much lateral movement occurred from the sidewalls of these tires. I'm not terribly hard to please when it comes to a car that only sees occasional use, but I'll be replacing these in the spring for any tire at all that doesn't make me feel like the whole back end of the car is coming loose.