let's assume you take 3month option on a Apple
You are deconstructing Rate in the call option formula from all other variables including option price. Since repo market is not explaining the difference I would assume that your are getting a premium over libor 3m rate. Assuming this premium is not justified by libor bid/offer spread.
First thing to check will be the forward price using call put parity. Can you back out the rate from call put parity instead? Ideally the risk free rate you get from the model should be lower than libor (Libor has credit risk embedded)