My default answer: build a new modern database, create a datapump to and from. Don't touch the old stuff, write any new stuff to access the modern database directly.
Inasmuch as new features require storing new data that doesn't fit in the old schema, just don't send that data back.
You get to implement new features piece by piece. But always have your fallback that --crufty or not-- keeps the company going. The big hairball in side-by-side systems is operations that require atomicity. Those can get hairy, so I'd ignore them, leaving that stuff in the old system as long as possible, until you're ready to rewrite those modules in the new system and sunset the old one altogether.
Inasmuch as new features require storing new data that doesn't fit in the old schema, just don't send that data back.
You get to implement new features piece by piece. But always have your fallback that --crufty or not-- keeps the company going. The big hairball in side-by-side systems is operations that require atomicity. Those can get hairy, so I'd ignore them, leaving that stuff in the old system as long as possible, until you're ready to rewrite those modules in the new system and sunset the old one altogether.