Nono... first you have an HC permanent galaxy, where one fam, say fam 7777, ends up dominating it and 'winning' that galaxy.  Instead of stopping the game at that point, you just let it keep going indefinitely.  That galaxy, Sagitarius, for example, becomes fam 7777's home turf, their galactic 'hood'.  They keep collecting resources and GC, fleet, etc.

Then you start another permanent HC galaxy, and same thing happens that round, except a different fam, say fam 7776, ends up dominating it.  Andromeda, for instance.  You keep starting galaxies, and, after a while, many fams end up having control of various galaxies, maybe multiple galaxies, although this is hard because their empire sizes makes it difficult and more expensive for fams who already have high NW due to controlling one galaxy already.  It's hard for them to be able to attack anyone unless they are attacked first, and they might be having to fight in their home turf galaxy, too, which would drain their morale everywhere else.  If they are getting attacked in their home galaxy by another fam, they can attack that fam in any other galaxy, regardless of its size or NW in that galaxy.  There are details to work out, but these can be worked out, I think.

The end scenario is that you have a number of fams (permanent fams, that is) who control permanent galaxies and interplay between galaxies.  For instance, one fam might make an incursion into one galaxy simply to dump their resources on that galaxy's market.  And all other fams try and make incursions into that galaxy to buy cheap resources (at least on of your fam members has to maintain a home planet galaxy in that in order to play that galaxy's market).  There could also be an inter-galactic market...

There's tons of stuff that could happen.

As I was saying, it would take IC to the next level.