What is this amplifier doing, exactly?

The 220 uF looks much better than 0.33 uF. The quick decay is gone, and you'll
not have any dc-currents through the speaker.

I thought the 100R in the emitter would help improve the low-freq response, but
it also kills the gain too much. So you might go back to tying the emitter to gnd,
and use a 10K in series with the 220 uF cap on the base. I might also use a larger R
on the base to gnd, eg go back to the 100K.

I'm not sure why you're getting the overshoot on the leading edge. Do you have the
scope probe ground lead tied close to the same point as the speaker gnd, or right
at the emitter?