Probably not. The infrastructure has to know where your cell phone is in order to communicate with it.

Last I checked, the large carriers in the states hoard this information for years.

This is not quite true. They have to know roughly where your cell phone is but, helpfully, your cellphone starts the process as soon as you start it up and then it is a courtesy to you that the unit closest to you will handle your call(s) and other traffic because that will save you battery and reduces the amount of power your phone will use which in turn will allow others relatively nearby to use the same slice of the spectrum while you are transmitting. Phased arrays on the mast make this even more precise and further conserve power. But that really is a courtesy: it would all work without that luxury but less efficient and your phone's battery would be empty faster.

The part that really is optional is where the carrier then stores and even sells your location. They are mandated by law with respect to the first and they abuse the technical capabilities of the system for the second. And even if it isn't very precise for a single measurement it is in fact quite precise after you haven't moved for a while.