I guess you need to filter the output. A high Q filter will boost the 388MHz fundamental as well as reducing the harmonics, allowing you to get a cleaner output for the same order filter.
Another way to reduce the distortion is to add the outputs from more than 1 of the inverters. E.g. if you have 5 inverters in the ring, you can add (or mix) the outputs from the 3'rd and 5'th inverters. This won't give a very big improvement, but it's simple and costs almost nothing.