The main difference compared to swapping the rectifier (ie 5V4) will be you will probably increase the heater voltage over the spec (6.3+/-10%) - not a great idea.
With either method, the 6V6 plates will run even hotter, and with a 5E3 they are pretty much at max anyway.
You could accomodate the additional voltage by increasing the value of the cathode resistor, ie 330ohms, but then you go further towards class B at high signal levels and crossover distortion may thin the tone down and go 'reedy'.
Higher B+ levels tend to give the 5E3 tone a harder edge - more like a mini marshall, losing the soft warm response.
If you want to stay clean but get louder, the most effective way will be to double up on the speakers, which, assuming that you keep the correct load impedance (and all else being equal), will give you another 3dB sound pressure level, which is equivilant to doubling the output power of the amp.