"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)?

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.

Ugh, if you must: “I can make time for a phone call, or exchange emails with whoever it is with whom you want to put me in contact”.

But I object to such unnecessary convolutions! Just say “I can make time for a phone call, or exchange emails with whoever you want to put me in contact with”.

There is no rule about ending sentences in prepositions, except, perhaps, among anal-retentive school-marms. It often makes for a lack of clarity and that should be avoided, but there are many situations where it actually makes the sentence clearer and more natural (and linguistics is descriptive and not prescriptive - language will evolve however it will and we have little to say in the matter, so watching actual usage is far less futile than making "rules").

But as to your question, I think you are simply suffering from an overabundance of words:

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

That sounds fine to me - clear and fluent. This is a case where the terminal preposition would be more awkward than its absence.

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

That sounds fine, too. It just shifts the focus, the burden of who is actually performing the act of making the contact choices, which is what I was avoiding in my construction. In yours, the burden is on your interlocutor. I kept it that way.