Maybe gear?
I never use hq materials (and don't know how to make them) and have no problems to HQ items with level up to 2 higher than my own. Also i have ~ 70% chance to get HQ at my-lvl+3 and 50% at my-lvl+4.

To do this i use modified 3-4star rotation with PbP*2 RS CS2*2 progress sequence. If item is very near to my level i replace one or both PbP to HT... If very far - one of CS2 to RS (risky!). Such rotations hits 12-14 HTs and often finishes quality progress even without hitting Byregot (for near level items).