When does a soldier get his bonus from extension on a six year contract? At the end of the 3rd year according to the ETS date?

As the quick blurb on your name states NG, it's spelled out in the NGB 600-7-R-E, Section III, P1:
"1. I am reenlisting/extending for the Six-Year DMOSQ REB and will receive a total amount of $XX,XXX. My incentive will be processed in two installments. The first 50 percent payment will be processed the day after my current ETS upon verification of my MOS and unit of assignment qualification in GIMS. The second 50 percent payment will be processed on the fourth year anniversary of my contract start date provided installment 1 was previously processed."
When hypothetical SM you are asking about signed the contract and bonus addendum, the full time staff or retention should have loaded it through RMS to the state, and it should be view-able in their iPerms.

in the real world.. a soldier gets their bonus when somebody gets around to actually pay it... Of course they forget about the taxes, and to see their gleeful little faces when they see the final amount.... The temper tantrums I've seen. Including my own.