This is a common issue. The negative output is from an emitter follower, and the base of that emitter follower is connected to the - rail through a 1K resistor. When the output of the amplifier slews toward the negative rail the voltage across that 1K resistor becomes less and less, and eventually will not be able to provide sufficient base current to the output transistor.
The trick is to use 'bootstrapping'. Coupling the output back to provide a negative voltage greater than the rail so that the output transistor is not deprived of base current.
Using your existing circuit, change that negative side 1K resistor into two 250 ohm resistors in series. Now add a 1000uF cap between the output and the junction of the two 250 ohm resistors. See if that doesn't improve you output range before clipping.
Also, it is good practice to use small value resistors in the emitters of each of the output transistors. Add a 1 ohm resistor between each output transistor emitter and the 'out' point.