Legal Question in Criminal Law in California

No. You cannot hire the attorney of your choice at taxpayer expense.

If you have a conflict with your public defender, you can make a Marsden motion. The judge will clear the courtroom so you don't reveal defense strategies and ask why you don't think your attorney is doing an adequate job. He will then give your attorney a chance to respond.

If the judge decides to appoint a different lawyer, it could be someone who can't get hired at the Public Defender's office.

Well, you could ask your Public Defender if his office has ever represented any of the witnesses who COULD be called against you if the case went to trial. If so, he has a conflict of interest and he should withdraw. Then the County appoints a private lawyer to represent you, and it IS paid at Gov expense. If he won't bother checking, and if you know any of their names, you can check the County's criminal index and find the case numbers and see from the court file who represented them.