How to work out the odd case?
If you have a solution for all even numbers, it might be sufficient to realize $T(k+1) \ge T(k)$.
But generally this is proved by induction where the even numbers use the odd ones. There, you have two leads :

Take good care of your bounds during your induction
More simple (but with less efficient approximation) : prove it for powers of two with a given constant ; and then use those to prove the general case for another constant