That was an excellent job of yours: the original axe is still there and it can be used again!!!
What I would recommend additionally is giving the axe head (and 1" of the handle) an extended soak in a jar filled with a mixture of boiled linseed and balsam turpentine (4:1).
Leave the axe in at least for a week, the let it dry.
This gives excellent protection to the wood against the too-dry air in our contemporary housing, a real killer for any handle. A slight coating left on the steel will also prevent rust when using it outside.
A rub-down with Ballistol protects the handle as well without making it slippery.
I do this once a year with all my axes.
Read here about the numerous possibilities of taking care of wood:
http://www.dick.biz/mailing/NaturalFinishing.pdf