Hello!
Is "available" at the right place, here? :
-Would you have another time slot available to suggest, please?
Thank you for your help
W
It's okay there. You could put it before "time slot" as well. You could also omit it altogether, since if the person is suggesting the time, it's probably available.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.
How about removing to suggest unless you are actually going to use that information to make a suggestion to someone else?