Your expected hours of care needed is what will hold you back more than your son's behaviour.

I usually try to give it three weeks before making any kind of decision. Aggressive behaviour is different as there is a responsibility to the other children to provide a safe place to play so that would need to show some sort of improvement within the first week or proof the child will listen and obey when reprimanded - reminded the behaviour is not acceptable at daycare. For the small amount she is going to earn she is likely not interested in even trying to work through the situation.

You say afternoon hours. Be sure your child is busy in the morning, has a good lunch and arrives in need of a nap. That is the only way the afternoons are going to work because providers guard that time carefully.

As far as what to look for in a new caregiver - make sure she has only 1 or 2 children and plans to keep her daycare very small so that your son can have the extra attention he will need. This may cost you more than a larger group setting.