Hi Alex,
I have no idea what SPDT is,
however i assume the reedswitch
is driven like this.
Sometimes the base of the transistor has
to be very low to turn it off,
especially if its a darlington pair.
If this is a mass produced item,
or sold as a working model, then this is
a failure rather than a design problem.
I assume this is a home made assembly ?
One way to get around the high base sensitivity
is to alter the circuit slightly to an
emitter-follower type.
The gain of the stage is still preserved,
but now the low voltage on the base
would comfortably switch off the reedswitch.
This may not be your problem,
but it might be.
Regards, John