What I am hearing is that the clingy behavior is an attempt to cope with some scary feelings. Rather than focusing on whether to indulge or discourage the behavior, I would work on the feelings.
Sounds like she is worried that life is, or is about to start, going by so fast she can't appreciate it. As her mind matures, she is able to remember farther back, and the distance between her 9th and 10th birthdays may seem much shorter than the distance between her 8th and 9th. She is better able to understand the sad things in life and may have a hard time distracting herself from them. As each day draws to a close, she's realizing that that particular day is gone forever; whatever the number of days in her life, there's one LESS still available to do all the things she wants to do.
Look for some ways for her to immortalize things she does. Does she have a diary? a camera? a scrapbook or memory box? Talk up how wonderful it will be all the rest of her life to look back at what she did when she was 10 and all the other exciting ages she's going to be. Look back at old photos and talk about how she's grown and how cool it is.
Talk about when you were 10 and show her whatever memoribilia you have. Explain how it's simultaneously fun to look back and admire who you were, and pleasing to see how you've grown and improved each year. Every age has its highlights. (Make sure she doesn't hear you fretting about gray hairs or other signs of aging.)
Find an important responsibility to keep her busy at the time of day when she tends to go into a slump. Is it possible she has low blood sugar in the late afternoon? Maybe she could help you make dinner and snack on stray veggies and things while working.
A book I loved at that age is Starring Sally J. Freedman as Herself
by Judy Blume. It's about a very imaginative 10-year-old girl who has many things to worry about but deals with some of them by imagining herself as the brave heroine in elaborate scenes. I'd recommend this as a book to read aloud to your daughter and talk about, rather than for her to read alone, because it has a lot of scary ideas in it but they could be great discussion starters. Another good book in that vein is The Lilith Summer
by Hadley Irwin, about a girl and an old lady discovering they have more in common than they thought.
Find a regular time when you can talk with her alone and really listen to what's on her mind. She'd like that to be at bedtime, but if that doesn't work for you maybe taking a walk together every evening would help to clear her mind before she lies down to go to sleep. It sounds like she needs some extra support and companionship sometime between late afternoon and falling asleep.
Now that I have some perspective on myself at that age and have been working with Girl Scouts that age, I am kind of annoyed that adults seem to attribute all emotional issues to puberty. I don't mean to pick on you or to say that hormones have NO role in this, but please remember: The brains of both girls and boys change around 11 years old as they move from the concrete operational stage of development to the formal operational stage; this is not a part of puberty but happens on its own timetable. The way their thinking WORKS is changing at a very fundamental level. That can make their old perceptions of things seem frighteningly naive and their newly grasped reality seem distressingly complex and perilous. They clearly remember being the Me of two years ago, but at the same time that Me seems like a different person, not just smaller and less experienced but fundamentally different. Some kids feel that transition as it's happening and are afraid to lose the old Me because they can't yet feel the new one.
On that note, one more idea: Try to celebrate every new skill your daughter learns. That should help to make growing up a more positive thing. If you or other adults have time to play games with her (without younger brothers), try some games that use more complex reasoning, like Zendo
, or Mastermind
I hope things get happier for both of you!