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Ben,

Perceived loudness is a factor of speaker efficiency and power delivered. However speaker efficiency has a larger weight in the equation over power.

1. Efficiency
For example, a very efficient speaker that is 95dB SPL (sound pressure level rated with 1 Watt of input power at 1 meter distance) will be extremely loud with 1 Watt of power. In fact, as the speaker efficiency spec suggests, it will be 95dB at 1 meter!
Now let’s take an 85dB efficient speaker. Every time we double the power we only gain +3dB in perceived loudness. So in order to get our 85dB speaker to be as loud as our 95dB speaker, we must gain 10dB, or 10/3= 3.33 factors of +3dB. So to find the power we must take 2^3.33 = 10 Watts! It would take around 10 Watts of power with an 85dB speaker to be as load as a 95dB speaker with 1 Watt.
This should convince you that if you want to go loud, change your speakers if you can.

2. Power
To get more power to your speaker you need to supply more drive voltage to it. If you amplifier has headroom in the maximum power supply voltage spec, you can increase the power supply voltage to get more clean signal swing and thus more power into the loudspeaker before hitting the power supply voltage rail limit at the amplififer output.