Replace amplifier IC with a class D amplifier (AN7133N with TPA3116 board)
If you want to socket the amplifier, then no. It's bridge output, not compatible with the existing speaker and headphone socket wiring, unless you leave the SPx- outputs unused, which reduces your output power by a factor of 4. Even with that restriction, it should still deliver more power than the existing. You should test the module for stability and distortion with one output unloaded however, some bridge output types may require a balanced load, others might manage without, check what yours does before committing much work to the project.

If you modify the speaker wiring to bridge, then you'll have more confidence that it will just work, and get the full power. The headphone socket will still need to be single ended though.