How to repeatedly read and write a SMD EEPROM without unsoldering it each time?
Since we're, either way, talking about a relatively intrusive change to your system, one could consider that


SOIC definitely isn't the smallest existing footprint for EEPROMS and
you essentially want in-system programming.


So, what about ditching the SOIC form factor EEPROMS completely? 

I imagine something like the "stamp" PCB format, ie. a small PCB with contacts at the edge that fits where your SOIC EEPROM would have been.

The only purpose of that board would be to


go where originally your SOIC EEPROM was,
carry an electrically compatible EEPROM,
allow to disconnect the EEPROM's power supply from the rest of your system, and
have test points that go to the address/data lines of your EEPROM


That way, you could power off your system, only power on the EEPROM (e.g. by pulling off a jumper and connecting an external voltage supply), and then use the EEPROM as if it was desoldered (assuming that your system has high impedance on the EEPROM wires when powered off).

You could permanently solder that PCB in place.

Notice that this often isn't even necessary. If you can build / buy a reliable pin probe, and can assure that nothing else accesses the EEPROM bus during readout/programming (e.g. by holding the microcontroller it connects to in reset), many typical buses (I²C, for example) will allow you to access your EEPROM while it's still built-in.