Practice playing memorized pieces "cold" (without warming up your hands); this is obviously more difficult than with warmed up hands but practicing under adverse conditions is one way of strengthening your ability to perform in public. This ability to just sit down and play cold, with an unfamiliar piano or environment, or just several times a day when you have a few minutes, is one of the most useful advantages of memorizing. And you can do this anywhere, away from home, when your music score may not be available. Practicing cold prepares you to play at a gathering, etc., without having to play Hanon for 15 minutes before you can perform. Playing cold is an ability that is surprisingly easily cultivated, although it may seem almost impossible at first. This is a good time to find those passages that are too difficult to play with cold hands and to practice how to slow down or simplify difficult sections. If you make a mistake or have a blackout, don't stop and backtrack, but try to keep at least the rhythm or melody going and play right through the mistake.
The first few bars of even the simplest pieces are often difficult to start cold, and will require extra practice, even if it is well memorized. Often, the more technically difficult beginnings are easier to start, so don't get caught flat footed by seemingly easy music. Clearly, it is important to practice the starts of all pieces cold. Of course, don't always start from the beginning; another advantage of memorizing is that you can play little snippets from anywhere in the piece, and you should always practice playing snippets (see section III.14, on "Performances and Recitals"). It is of course wise to memorize the beginning well. What are the key/time signatures? What is the first note and its absolute pitch?