Dwarf Everbearing can fruit alone, but it still benefits from a partner. Named fruiting mulberries often fruit alone, but mixed plantings can improve harvest size and give you a longer season. It flowers in mid bloom, which is worth knowing if you add a second tree later. Usually grown in USDA zones 5-10.
No single variety stands out as the obvious partner, but that is not a problem here: Hicks Everbearing, Illinois Everbearing, Kokuso, Persian, and Wellington all bloom close enough to work. There are 2 more in the list below. Pick whichever your nursery actually carries. Mulberries are often wind-pollinated, so close spacing helps but they do not need to be packed as tightly as pawpaws or blueberries.
One thing to keep in mind. Named fruiting mulberries are usually the safest picks. Seedling mulberries can be male and never fruit.
To put that into practice: at a minimum, one named self-fruitful mulberry is usually enough for home harvests. Better still, plant two named fruiting mulberries if you want more fruit, a longer harvest, and a backup if one tree fruits lightly. If you already have an unknown seedling mulberry, wait to confirm it fruits before counting on it as your only tree.