I think your idea of using more than one site in a unit cell is the right thing to do for creating multi-component / multi-band models of interacting electrons. Basically let's say you have 3 "flavors" or "labels" of electrons. Then sites 1,4,7,10 etc. are all considered (by you) to be flavor 1; sites 2,5,8, etc. flavor 2; and 3,6,9, etc. flavor 3.
You then just define the Hamiltonian parameters as from on paper, so like if t1 is the hopping between sites which are both flavor 1, you just have t1 go between every third site, being sites 1,4,7, (i.e. the flavor 1 sites). That is it's a 3rd neighbor hopping. Similar for a t2.
In contrast, a local term like a Hunds interaction would happen inside the unit cell, so like between just sites 2 and 3, then sites 5 and 6, etc. repeating that way.
If that's still not too clear, perhaps you could provide the specific Hamiltonian or a link to it and we could discuss more?