Paying a dollar to reflip four coins

Four fair coins are tossed and you are paid $1 for each head. After seeing the result you may, once, pay $1 to toss all four coins again, in which case the second toss determines your payout. Your rule is to retoss exactly when the first toss shows $0$ or $1$ heads. What is the expected value of the game under this rule?

$\tfrac{33}{16}$ dollars. Keeping 2, 3 or 4 heads contributes $\tfrac{2\cdot6+3\cdot4+4\cdot1}{16}=\tfrac{28}{16}$. With probability $\tfrac5{16}$ you retoss, earning an expected $2$ minus the $1 fee, adding $\tfrac5{16}$. Total: $\tfrac{33}{16}$.