Gulliver is an adventurer at heart, and it is doubtful to the reader that he would be so meek as to abide by the mandates of his wife and keep house with the family.
At around 10 days after returning from the second voyage, Gulliver says that Captain Robinson, commander of the Hopewell, visited him.
As he asks Gulliver whether he was "set up for life", meaning, whether he felt that he was going to stay at home for good, he spoke about a trip to the "East Indies", and that he wanted Gulliver to be surgeon of the ship.
Robinson's offer was quite generous, and his treatment of Gulliver was kind and familiar. Moreover, Gulliver was not just going as a passenger; he would lead and put his skills to good use. The money was going to be good, too.
I should have another surgeon under me, beside our two mates; [...]my salary should be double to the usual pay
Moreover, the captain gives praise to Gulliver, stating that the latter's experience at sea is just as good as the Captain's, therefore, he would be obliged to obey Gulliver's commands and advice as a consultant, essentially.
...having experienced my knowledge in sea-affairs to be at least equal to his, he would enter into any engagement to follow my advice, as much as if I had shared in the command.
We learn that Gulliver's wife eventually consents to this trip, explaining to the children the financial benefits that were to be obtained from it. This, and arguably Gulliver's own hunger for adventure, were the pivotal reasons why he ends up going on a third trip.