OK, if you are not leaving the country (and if you are, using both passports is OK for that), if you are not leaving the country, at some point your 90-day report 'obligation' will be due.
Do you do your 90-dayers online? In person? Or by mail?
This will affect the decision when to get the extension stamp (and re-entry stamp, should you possess one in the now-expired passport) transferred to the new passport.
Doing it online and using the new passport number won't work, you'll be rejected and asked to report in person. (You might try using the expired PP number, you would probably be approved, if the online 90-day reporting system's been working for you. But you also might be discovered later for having submitted a PP number that was not valid; don't know what the repercussions might be, but maybe wouldn't want to find out.)
There has been at least one report, maybe a second, of successful first-time-after-a-new-passport 90-day reports done by mail. Imagine that is because physical copies of the stamps and new PP data page are sent to the immigration office. Online, all they are seeing is a number and not the new passport front page.
If you do a 90-day report in person, either by choice or being told to do so after a rejection notice from an online submission, you will have to have the stamps transferred to the new passport before being allowed to do the 90-day report.
That is probably the ideal time, if not mandatory, to do the stamp(s) transfer, when doing your first 90-day report--if you are obligated to do one--before your next extension.
Having done a stamps transfer at CW within the last year, doing the transfers and 90-day report independent of any extension, I'm glad it went that way. If one deals with an office not so busy or active, doing the whole hog in one go might be OK. But there's enough to do for just a stamps transfer. And it worked out good to get all that cleared before extension time.