Plenty of nice anchoring spots in North Sound. Just west of Vixen Point at Prickly Pear Island has excellent holding, not too deep, and protected from E to NE winds plus you don't lose the sun behind VG as early as at Leverick.

The Spanishtown anchorage & moorings is a good option as mentioned if you're looking for some dinner ashore.

You also have the anchorage in Savannah Bay, VG in settled conditions and just south of Mountain Point, VG.

I've sailed to Anegada from St John via Soper's Hole, but you need the wind to be well to the SE for a one-tack sail, which you can get in June. Jost Van Dyke, Guana Island, and Cane Garden Bay are all possibilities with the prevalent SE wind in June if you look forward to a 4 hr sail.