How specific to be in headhunter into call for PE off-cycle?

I have a HH call scheduled with a firm that has pretty wide reach to a lot of reputable UMM/MFs and I'm wondering how specific I should be in my interests for firm size and type? Some info for context would be that I didn't engage in on-cycle due to personal matters that were out of my control, so I'm going through off-cycle for 2025 placement, which from my understanding of reading through WSO and word of mouth seems a lot of firms still have open spots/haven't fully kicked off recruiting yet.

I know that it's obviously best to be very clear and specific with your intentions and goals when talking to headhunters and the more specific you are the better, so I know the type of firm and strategies I'm interested in, but is it ok to say that I'd like to target both UMMs and MFs? My bank/group doesn't have the best placement into MF but still does place occasionally. It places better into UMM, but I wouldn't want to close off MF options to me by saying one or the other and want to tell the HH that "I'm interested in firms ranging from UMM to MF in size." Is this considered not specific enough and should I just stick to one of the two options then list off shops in that interest me there?

Also, I am not really interested in MM PE (I apologize if this makes me sound like a douche as that is not my intention). I have interned/worked in MM PE for a majority of my undergrad and would like to do something on the larger scale since the MM didn't fit my long-term interests.

Thank you in advance for any help!