The C5 bootstrap capacitor performs two functions.
a) it raises T3 collector load to increase its gain
b) it provides extra voltage headroom on R9 to drive the output pair
T3 gain is limited by the collector load, here being R8 and R9. Ideally the collector load will be infinite. However, the T3 amplifier needs a fairly high current in the collector to source a decent current into the output pair, which would require very low values for R8 and R9.
One way to provide that would be to replace R8/9 with a current source.
This amplifier designer has chosen an different way. Without C5, R8/9 would simply present an impedance of 2.2k. However, C5 connects to the output of the amplifier, whose voltage follows that of T3 collector. C5 drives the mid point of R8/9, so that the voltage across R9 is now constant, at least at AC frequencies where the impedance of C5 is small. If the voltage across the resistor is constant, it's behaving as a current source.
When large positive outputs are required from the amplifier, C5 can drive the R8/9 junction above the rail, to continue to supply sufficient current into the output pair through R9.
The arrangement where AC-coupled feedback from a voltage follower is taken back into a bias arrangement to raise its effective impedance is called bootstrapping, and is also often found raising the impedance of input bias resistors at emitter follower or op-amp buffer inputs.
The arrangement where an output drives a capacitor to provide bias voltages above the normal rail is also called bootstrapping, and is also found in high side N channel FET drivers, to provide gate voltages above the normal positive rail.
This circuit qualifies for both types of bootstrap.