[ThrowawaySR]

My memories from flashing phones circa 2003-2004 may be a bit rusty (btw. back than "flashing" meant something completely different than nowadays ROM uploads, it was more akin to poking machine code memory locations in BASIC on 8-bits :) ), but if I recall correctly, the device periodically scans all the BTSs it can see and sends that list with signal strengths to the network (and the network decides which tower should the device connect to!). Something tells me that the device even stayed connected to multiple strongest BTSs at once, but that may be a false memory.

Of course, that was old GSM days, it may be something completely different today in 3G/4G/LTE, but so is on the other hand the location hardware and algorithms of carriers.

But no, no handoffs were needed to track you very precisely even back then. There were ways around, but given the poor opsec in the case, it is doubtful something like that was used here.