Are there any pitfalls to be wary of when purchasing an "umbrella" insurance policy?
An umbrella policy is insurance on top of insurance in case your primary insurance is not enough. For this reason it is usually inexpensive relative to to other insurance policies. 

Using your example, a million dollars beyond what you are already insured for is a pretty good amount of coverage. It would take a really catastrophic event for a small plane to do more than a million dollars worth of damage beyond the limits of an owner's or renter's insurance policy.

If you're considering an umbrella policy be sure that it does not specifically exclude aircraft because many do. If you purchase a policy through an agent who knows that you are a pilot, you're probably fine but don't assume that your commercial umbrella policy covers you because many commercial umbrella polices specifically exclude aircraft.
I would be more worried about the umbrella policy excluding aircraft (like you mentioned unforeseen things rendering the policy ineffective in the last sentence of the first paragraph) rather than a million dollars on top the normal coverage not being enough.

I'm also assuming that by typical pilot you mean small aircraft not a corporate jet or something that has a lot more potential liability.