Hello.
After an LP, you should have 2-4 hours of rest lying down supine. This itself reduces the chances of further complications. The second best option is to lie down while you are traveling back home. You should avoid sitting up. It has been observed that those who can manage to take a 204 hours rest after the LP have minimum side effects.
Regards
While I'm no medical expert, I've also heard that lying flat after this procedure will help prevent a headache. If you can have someone drive you to the procedure and then home, you can lie in the car and then obviously lie down when you get home. I'm sure this all seems obvious to you. Good luck.