I don't know what socket is used here, but on my 2 boards with AUGAT sockets (pin contacts are colet type, same as high-quality DIP panels) there is no visible difference in tempco, due to socket.
Compensation resistor can help to improve tempco from 0.1-0.2ish ppm/K to <0.05 ppm/K. If you see 1ppm/K, something is very much off, so need to fix that first.
in the R4/R5 divider backwards.
Jeez, If that's the LTZ which is in board now, that likely caused problem for LTZ too. It takes seconds for A-chip to reach setpoint temperature due to 400 C/W thermal insulation between little die and metal can.
PSU need to have high isolation from mains and avoid ground loops. It's easy to check if you have power issues: connect LTZ output to meter in normal polarity, record reading. Then reverse output connetion to connect LTZ output in reverse polarity. Absolute voltage level should not change. If it changes, you have power problem or parasitic ground loop that steals current.