At my Daughter's school, a public school... when the Kinder kids start, everyone... is super observant of them and helpful.
They, have a Counselor, for the younger kids. And, if a child has a hard time adjusting (my daughter was one of them), then, the child meets with the Counselor... and what they do is not "Counseling" per say... but play. They let the child express themselves, anything under the sun, and they talk interactively about it and just play. The Counselor, 'helps' the child by being someone the child can go to for anything... even if just for a hug and compassion. Anytime during the day. The Counselor also then serves as a "Buddy" and someone the child can rely on... when they need extra solace, because their Mommy is not there. The Counselor also does fun activities and crafts projects with the child... it is the child's "special" time. JUST for her/him. The session is about 1/2 hour, daily. And it does not conflict with the classroom as it is held during an open period or during recess. The child sees the Counselor for about 1 month. Or longer if need be.
And NO, it is not a 'stigma' nor do other kids think bad of that child... in fact, most of the kids... also wanted to see the Counselor because it is so much fun... and so 'special.' My Daughter, felt "Special" going to her Counselor. It was looked at very positively, by herself and the other kids and Teachers. Because, teachers do not have 'time' to do this with each child themselves.
THIS totally helped my Daughter. Who went through what you are describing. She LOVED LOVED LOVED her Counselor. This Counselor is ONLY for the young children, the Kinder and 1st Graders Jr.K kids. AND she also actively visits the classrooms to observe the "new" Kinder kids.... to make sure the children are adjusting well.
The kids LOVE this program... and it helps them adjust, to school/Kinder.
What your daughter is going through is not unusual.
Talk with her Teacher, or school Counselor.
It is growing-pains for the child... and per their emotional maturity or ability to adapt. Each child being different.
But do talk with your girl. And let her tell you anything she feels about school or her Teacher... ESPECIALLY if there are problems/peer problems/kids not treating her nicely/bullying, etc. Because THOSE are the things... you can bring up with the Teacher. We as Moms, need to 'advocate' for our child, when problems occur.
I ALWAYS talk with my daughter about school/her friends/the Teacher/the interactions/recess time/who her friends are etc. We talk story... I don't make it an 'inquisition' but rather... make it a fun talk story time with her... and we also bond that way... and she knows as a result, that she CAN talk to me about school and her feelings on it... about anything... because I "allow" her to do that.... and a child needs to 'vent' too. And that is ALSO how I 'gauge' my daughter and her well-being, in school.
all the best,
Susan