5.5 mm pellets with thin skirts work OK in alot of old 5.6 mm barrels. They must blow out and grip the rifling. Try RWS Superdome.
He’s not advertising them any more so Steve Z may well be right. (Glad I stocked up a few years ago.)
Superpoints also have nice thin soft skirts and reportedly work well in 5.6 barrels. I wouldn’t normally recommend a pointed pellet for any purpose, but they are inexpensive and worth trying.
Milbro is now an SMK brand for things made in China but I think pre-SMK Milbro .22”s are on the large side too, even if, like Marksmans, they aren’t exactly JSB quality.
Talking of which, JSB offer a 5.53mm head size. Take quoted head sizes on any pellet with a pinch of salt, but they could be worth a punt too.
I have got adequate to excellent results with RWS Superfields in older 5.6mm springers, but I appear to be in a minority of one on that, judging by forum discussions.