should we assume this is a ring oscillator on a semiconductor chip? i.e. not discrete?
if i were experimenting to get better phase noise, i would first look at the capacitor. the capacitor adds some time delay, which can be used to "tune" the frequency. but some capacitors are not really linear capacitors. Since you are putting a large signal voltage on this cap, with all sorts of harmonics, its C vs voltage response becomes important. if the capacitor is in fact non-linear with voltage, all those harmonics can fold back into the baseband oscillating frequency, and appear as noise (half amplitude noise, half phase noise). but since it is a digital circuit, and there are possibly voltage limiters in the circuitry, any AM noise converts pretty readily to phase noise (or time jitter).
I WOULD make sure the power supply is clean. i would be tempted to run it off of a linear voltage regulator, rather than a switching regulator. and like said, you need multiple bypass caps...like a 10pf in parallel with a 100 pf, in parallel with 0.1 uf, in parallel with 10 uF. probably ceramic for all of them.
I would also make sure the load is impedance buffered. if the load impedance the ring oscillator sees is changing (even if only by a minute amount) that will cause phase noise too. i would put a bunch of buffer gates in series at the output of the oscillator.