Non-RC electric wiring questions

I like to tinker with electronics, but I do so mostly by dissecting things and making sure before I just a wire I know where it goes so that when I rewire I can make it work (aka I don't actually know what I am doing!).
I took an old fm/tape walkman apart because I wanted to dissect the AM/FM radio part of it to make a new project. I want to use that am/fm radio circuit board and attach it to an old speaker in discrete box so that I can have a radio playing without having a radio out in the open (I like to confuse people which I am sure I am already doing on this thread).
I wiring everything up and it works, but the speaker is not very loud. The speaker is rated for 40V and I am only putting 3V through the system (2-AAs).
Before I took the radio apart it was only running on 3V so I assume that the wiring and board shouldn't be pushed to a higher voltage.
So finally to my question!!!:
How can I increase the voltage (thus volume of the speaker) and not fry the circuit board?
Can I just wire more voltage in series to the speaker?
I understand basic wiring, but when you start getting lore then 5 components on a circuit board I get lost.
If you can give me the answer, or point me to a good tutorial that would be great.

Here is a question for all of you Chinese knockoff gurus:
Any idea where to find a pre-assembled am/fm radio circuit board setup for ~3-6v DC for cheap? I can disassemble an $8 radio if there isn’t anything cheaper.

Thanks for your help!