grammar - "Exchange emails with whomever you want to put me in contact [with]"

I realize the "never end a sentence with a preposition" rule is controversial these days, but let's assume for the sake of argument that it should be followed. What is the proper construction of a sentence that has a prepositional phrase inside a prepositional phrase, such as in the title of this post?

Is there a way to make this strictly correct, but not so cumbersome?

I can make time for a phone call, or just exchange emails with whomever you want with whom to put me in contact.

Is this simply a case where there is no way to avoid ending the sentence with a preposition (short of clobbering it with the ugly stick)?

Answer

I can make time for a phone call, or
just exchange emails with whomever
you want me to contact.

But seriously, yer killin' me with the preposition thing.