To repeat my comment: if you are looking for ways to help students to break into already-formed groups without awkwardness, then I think you are trying to solve a problem that is bigger than just your office hours!
That said, there are strategies I use to get students working together in my Maths Learning Centre and in tutorial classes.
I find that grouping people together works best when everyone in the group is still working on the problem. It never works nearly as well when someone has already finished it and is helping the others because the others are in "receiving mode". So perhaps look out for more times when everyone in the group is struggling with the same problem, and encourage them to work together. You might need to stay to help them ask each other useful questions, but it seems to work better than the alternative.
If you do send a more experienced student to help a group, then remember that it is most likely that the more experienced student will be acting in the teacher role, regardless of your original goal. Therefore what you need to do is help that student to get better at passing on what they know. I find the best way to do this is to go with them. The experienced student knows they have you to fall back on if they mess it up, and you can support the other students to respond appropriately and offer more of their own thoughts, and you can help the experienced student to stay or leave when the explanation is over (thus avoiding the awkwardness). Finally, being there lets them know that you really do care about students helping each other.
You can get groups working together at blackboard/whiteboards. In my Maths Learning Centre, the moment anyone starts writing on a whiteboard, students will drift towards it to join in. Because working on a board is more public, people are usually more inclined to ask questions and offer opinions. It also has the advantage that they can change what's written there more easily and so construct a better solution than if they were writing on paper.
Finally, you need to set the ground rules. In my first tutorial, I would tell the students we would be working in groups at whiteboards because they needed practice at solving problems where they could get advice as they were working. I would tell them that they were expected to ask each other questions if they weren't sure, and to explain what they were doing as much as possible, and especially if anyone asks. I also told them that there were no stupid questions, but that a specific question like "how did you get from this line to this line?" is better than one like "What's going on?"
So in summary:
- Get people to work on problems they all haven't finished.
- Help more experienced students to be better teachers.
- Stay to support the interaction.
- Get people to work at whiteboards/blackboards.
- Set ground rules.