[iykwimthrowaway]

Another alternative could be what my company did for me: they gave me special permission to sell some number of my shares to one of their investors, who were always hungry for more shares. This allowed me to exercise all of my options, then sell off just enough to cover my tax consequences.

This seemed like an extremely good-faith maneuver on my company's part: they acknowledged that my options were a reward for past work, and they didn't want me to miss out on the benefit of those just due to some tax-related time pressure. They also knew it was really no skin off their nose if the investors got to acquire more shares at a price they deemed fair.