You also run into issues of what do with the wings on the way up. You can't just put huge ass wings on that thing. You likely need it deploy-able.

And then the wings would also survive the flip and vertical landing. Or if you want to land like a plane, then you also need landing gear.

So there is really no way to add wings without adding a huge amount of mass. You are building a completely new thing.

There are some super cool mega-space planes designed in the 70s (I think). But of course these were never built or even tested. I remember they had some overlapping metal heat shields and a big ass delta wing. They would also start vertically and use air breathing engines.