Children are at school to be educated and from your OP it appears you are more concerned about your child's social life and not her learning

Obviously learning is the main purpose of school but we are talking about children here, for whom the knock on effect of being isolated and unhappy will often be a downturn in their learning - it's not rocket science.
You have my sympathy OP, and other PP's in similar circumstances. My DD is going into Yr 6 this week and turns 11 very soon. After her lost solid friendship drifted apart at the beginning of Year 3, she made some other friends, both boys and girls but none of them really 'stuck', In Year 4, despite and episode of bullying she managed to acquire 2 male friends, who are lovely and great fun but...one has just changed school and the other has been put in a different class and will be moving overseas at the end of the year.
They have mixed up the classes as well as changing teachers for the third year running and gone from 3 days transition at the end of term, to just 2 hours in their new class at the beginning of July
. There are girls she knows in the class and is on good terms with but they have known each other since pre-school and are an established group.
DD is absolutely distraught - I've had her up until midnight in floods of tears, saying that she hates school, can't wait to leave for secondary next year and there's no point working hard for her SAT's because she's so upset she'll do it all wrong anyway (probably a touch of the drama llama there admittedly!) - I know my daughter (arguably much better than the SMT who made these decisions as they've only been there a year following academisation) and this is not her at all.
However, I'm not asking for her to be moved at the moment as she has a great teacher and I think a little space between her and her best friends before he goes away may make it easier for her in the long run. I just hope it pays off as the school seem to have a similar attitude to the HT quoted above all about the results and box ticking, never mind that there's small humans involved, and my requests for a little extra support with transition for her have so far been largely disregarded.
Sorry OP - that was a huge rant - it's weighing heavily at the moment with school starting Thursday and I did have a rant on the school's website last night after they posted a request for parents to 'model a positive attitude to the changes as this can be an anxious time for the children'
.
FWIW I would say there is absolutely no harm in asking, but be prepared for a no. Outside clubs are a good idea, as not only might she meet different children but she may be able to socialise with some of the girls in her class in smaller quantities (i.e. if just a couple of them go to Brownies) and find her way into the group that way if that is what she wants. Don't discount the boys either - DD's male friends are really awesome, although she does occasionally need a girl to confide in now she's a bit older.
I hope you managed to get it sorted and sorry for the hijack! 