Tate and Scout are both quite unisex but I guess Tate is you child already. I would use something a bit more feminine for a girl though Tate and Scout have the same feel. I love Tate and Tess mentioned by bluejuniper and I think Clarinda is too long in comparison with Tate.
My first thought was Tate and Sue but I guess it's not your style. Other ideas:
India nn Indy
Lydia (from your signature)
Aspen (espeially love this with Tate)
Or Alexis, Phoenix, Sasha if you like unisex names.