How can I get my newly clingy teen to feel more secure without me?

There are a few details to consider in answering this question. How old is your teen? Has she always been attached to you? Is this behavior totally new, or an exaggeration of a pattern of clinginess?

Assuming that this is a significant change in her behavior, what is the reason for this change? There most likely is something happening in her life that is causing her to regress and have a need for very basic, physical comfort and security. Look at her present life circumstances. Has she experienced any recent losses? Tragedies? Changes of her own or family situation? Is she in high school and beginning to be fearful of going away to college? Has she experienced some kind of trauma?

At this point, don't be too hasty to push her away. She may very well need you to provide security and pushing her away could make her more anxious. Talk to her and try to find out what's happening in her life. If you can find a major source of stress, then there is the starting point for conversation. Once you understand the behavior, then you can begin to determine how to help her. And if she won't open up to you, find a reputable psychologist specializing in teens to gain insight and guidance.