Chaining output of amplifier (to speaker) to another amplifier
Yes. you can do that so long as it's an 8 Ohm dynamic speaker and not something weird like an electrostatic or a piezo. The way you have described the connection is correct. If you happen to get a lot of distortion when you do this, it means that it was being driven by a class D amplifier. Solve that by adding a 0.5W 8 Ohm resistor in parallel (your new amp will have to do way more work if this is the case, but that shouldn't be untenable).