I wouldn't call this a Minhag in the halachic sense of the word (TBD).
This is more like a shul operational procedure.
E.g.: The typical shul will use whatever Paroches they have, all year round.
Then somebody will donate a nicer one and they may use it for Shabbat, switching it weekly.
Then somebody (who knows that the we need a white one for the High Holiday, else the services are invalid, or at least defective) will donate a white one and they'll use it for the High Holidays.
If somebody then decides to donate a 4th one they may use it for Chag, removing the white one after Yom Kippour.
Older shuls often have a huge collection (like Breuers) and they will decide that one white one is for Rosh Hashana, one for Shabbat Shuva and one for Yom Kippour. The purple ones for Pessach. The nicer green one for first days Sukkoth, the others for Chol HaMoed, the old white one for Hoshana Rabba, and the colorful one for Simchas Torah, etc.
As a new Paroches is donated, the rules change.
Similarly, which lights and chandeliers get lit when (weekdays, Shabbat, Rosh Chodesh, etc.) is more of a procedure than a Minhag. Ditto for which Silver Torah Ornaments to use when.
After many years, the kids notice a fixed pattern and are convinced it Sinaic, or at least an ancient Minhag. When they grow up they then teach their kids the significance of this Minhag and they wonder why other shuls don't follow their Paroches-Lighting-Silver Minhag - and of course they will donate the missing ones to their favorite shul.
But I think it remains a simple Shul Operation Procedure.
So, back to your question: When to change the white look? that depends on the size and colors of your Paroches collection, and the procedures the Gabbai/Shamash/Busybody who does the switching has in his head, aka (mistakenly) the local Minhag.
Source: This user's experience who grew up with these very important Minhagim and has been inventing these Minhagim locally for over 20 years as our shul grows, matures and gets wealthier.