cpotter, if you notice on the schematic that you included of your generator, you can see that in the generator you have what looks like a primary transformer winding and two secondary transformer windings. If you look closely at the secondaries, you will see numbers representing the wires leading from the windings.
You will see that there are 4 wires, one being labeled #3, one is labeled #14, one is labeled #9 and the last one is labeled #5.
Your generator has two 120 volt output windings. They are connected in series so that you can get 240 volts out of your generator. Your generator provides a connection to the midpoint where those two windings are connected together, that point becomes effectively the neutral of the 120 volt outlet sockets.
As you can see, #14 and #9 are connected together. Together, they become effectively the midpoint between the two hot output wires, #3 and #5.
In operation, while #3 goes positive 120 volts compared to (9 and 14 together) your #5 is going negative compared to (9 and 14 together) by another 120 volts. Number 3 and number 5 are both exactly 120 volts away from the combination of 9 and 14, but, they are out of phase with each other, while one goes negative, the other goes positive.
Those voltages add together because the windings are in series with each other, so, the voltage between 3 and 5 is 240 volts.
Your generator is giving you two choices, you can connect a 120 volt load to one of the windings or the other, by choosing one or the other of the 120 volt output sockets. But, you cannot get the entire generator's output, only one half of the generator's output, because you can only connect to one winding at a time.
Now, to answer your question, if you could modify your generator, completely unhooking all four wires from the rest of the circuit, you could do this: Connect wires #3 and #9 together and label the combination L1 just for grins. Then, connect #14 and #5 together and again just for grins, label that connection as L2.
If you measure the voltage between the new L1 and L2 you will find your 120 volts and you will have the entire output of your generator available.
There were many, many generators built from the factory having a nice big switch on the front panel that could do exactly this for you.