Right hip/ glute med tightness

I’ve been experiencing some pain in my right glute med that I can’t quite seem to pinpoint. It almost feels like it radiates from my lower back on the right side and is like a constant dull pain. It’s the worst in the mornings when I first wake up and sometimes when I’ve been sitting for prolonged periods of time then get up to walk. It does not hurt at all when I run and actually feels better when I’m moving / exercising and warmed up— which to me seems like characteristics of a tendon but again I can’t pinpoint it. Certain stretches aggregate it like hamstring stretches if I go too aggressively. But light “pulsing” stretches feel ok. Again, does not bother me when I run, more so in the mornings and night when I haven’t been moving that much. No idea if it’s actually stemming from the glute med, or if it’s like tight hip flexors/ psoas pulling on something back there, or possibly a slight sciatic pinch??

Any suggestions for relief or thoughts about what this could be is much appreciated.