You can monitor the voltage across the switch using a comparator.
Add a bias resistor from the +12V to the left side of the switch that would give give a small voltage (say a little over 0.5V) across the relay coil on the right (resistor value about 20 times the coil resistance). This will not affect relay operation.
Monitor the right side of the switch with a comparator (such as one in an LM139 quad comparator).
Connect 12V and ground to power the comparator.
Connect 0.5V from a voltage divider to the comparator negative input (24k from 12V to the negative input and 1k from the negative input to common).
Connect the positive input from the comparator to the right side of the switch.
Connect the comparator output to a 10k resistor that goes to +12V.
When the comparator input is at 0V the comparator output is low (0V) indicating the switch is open.
When the comparator input is above 0.5V (either from the bias resistor or +12V) the comparator output is high (+12V) indicating the switch is closed.