Replacing socket, only one hot wire, two neutrals from different places
Thanks to Harper's comment on my question, and his comment on a deleted answer, I came up with a solution.

Both to play it safe (i don't know why, but I don't like the idea of using a socket as a wire nut) and because I did not have enough length left on the original hotwire, I went with the following:

I tied the two original neutrals as well as a new neutral together with a wire nut. The new neutral goes to the top neutral side of the socket.

I tied a new hot wire to the original with a wire nut, running the new hotwire to the bottom hot side of the socket.

No tabs broken on the socket.

This seems to work fine.