Even LIBOR isn't a great measure of the real funding rates at the short end, for reasons discussed in another thread.
Over here we use Reuters df's for this purpose. Pretty much everyone uses futures for the middle of the curve and par swap rates at the end. Reuters uses both LIBOR and quoted short-term deposit rates at the short end. In the current environment, this makes the Reuters curves much more volatile there, and the rates are generally higher than LIBOR.
It makes sense to use these since what matters for hedging, risk, capital allocation and so on is the rate at which *you* can fund, rather than the rate at which, say, Treasury or Goldman Sachs can.
...And this raises a whole series of issues, since the rate at which you can fund and the risk-free yield you can get in the market are undoubtedly not the same. So even an instrument as simple as a forward contract gets more complicated: For you, the spread in rates means that there's a long forward price and a short forward price, and the two aren't the same. Following this through to put-call parity and dynamic hedging arguments, this means there's one forward for you for long call / short put positions, and a different one for short call / long put positions....