If you think you're pressing too hard with your thumb, that's probably the issue. Try playing for instance a scale without any support from the thumb, use your right arm instead as a support. This is of course not how you should play, but should show you how little force actually is needed to press the strings.
Try lowering the action of your bass too to reduce the force necessary to press the strings. If you don't know how to do that, check the setup and repair forum. There is probably all answers on that.
And... keep practicing that spider.

It's good for the coordination and will build more strength in your fingers.