i think you dont want to use a amp and use the box only.
best option Change the speakers.
Go for free edge speaker to get more bass.(they move a lot)
or You can also go for a little bigger magnet for better fidelity.
Most boombox have enough power to play bigger speakers than provided.
Even a pocket radio like Pansonic RF-P50 can play 6 inch speaker. i used to do it ,belive me.
Most boomboxes are fitted with low wattage speakers just to make it loud( actually its just distortion).
A speaker need support from the cabinet to manage the bass.
Dont know if the box(cabinet) you are talking about is supporting the speakers or not when you compare it with a 3 piecer which has seperate boxes for the speaker. Sometimes a boombox cabinet doesnt do full justice to the inbuilt speakers. You can experiment by removing them and playing in a different speaker box(wood preferably).
so here are your options.
- If the box (cabinet) makes the best use of the speakers go for better speaker that fit there or refoam the existing with free edge ones.
- Go for car speakers they are the best you can even use old ones
- attach external speakers on the box like from a mini sytem. ((works best)) or make your own. Dont go for heavy stuff.
caution: but please follow the limits of your boombox. Dont go for speakers that are too heavy for the system or you will fry it. also the supported ohms of the box is important while choosing the speaker or you wont get loud sound at all.
Experiment. You will be surprised for sure.