If you are not card counting, the house has its edge/hand no matter what. For N hands, you just multiply the house edge by N to get the new house edge.
If you are card counting, the question gets more complicated, but one additional hand is not likely to make a difference between favoring the player and favoring the house. Any difference in player edge is going to come from the fact that more cards are dealt out between hands (card counters get most of their edge from changing their bet sizes based on the distribution of cards left in the deck).
All of this assumes that dealers reshuffle well before reaching the end of the deck/shoe, which is the case virtually everywhere these days.
By the way, the assumption that no one lets you play one hand at a table by yourself is false. I did it all the time when I lived in Washington state.