The former Germany international recently signed a contract until January at Goodison Park following a successful trial on Merseyside.
"First of all my focus is on the short term," the 30-year-old told the Toffees' official website.
"There are not many games between now and January but I want to stay beyond January because I enjoy it so much here.
"I think it is in my hands. I will get lots of opportunities to show what I can do in training and hopefully I will get plenty of games and I can prove to the gaffer I am worth keeping here beyond January. That is why I will work very, very hard."
Hitzlsperger was an unused substitute for Everton during Sunday's 1-1 draw at Queens Park Rangers.