No. Logically, your consent to their recording is not the same as their consent to your recording. It's a question of who has the right to make a copy. If you don't want them to record you, stop talking as otherwise you have given implied consent. If they don't want you to record them, you first have to tell them you're recording, so as to obtain [or infer] the necessary consent.
By comparison, if you're sitting face-to-face and whip out your recorder when they whip out theirs, that's clearly not a "secret" recording by either of you. If, however, your recorder is hidden in your purse, and not all parties consent to being recorded, that would be a felony in some states. By analog, unless the CSR in the OP's scenario knows of the second recording being made by someone else on the line, making it would almost certainly be a violation.
Sample of such a law: MA General Law c. 272 s. 99, in part:
(B)(4) The term ''interception'' means to secretly hear, secretly record, or
aid another to secretly hear or secretly record the contents of any
wire or oral communication through the use of any intercepting device
by any person other than a person given prior authority by all parties
to such communication;
(B)(3) The term ''intercepting device'' means any device or apparatus which
is capable of transmitting, receiving, amplifying, or recording a wire
or oral communication... (not including hearing aids or telephone
instruments provided by or being used by a common carrier).
There are limited exemptions for authorized law enforcement actions, among others not directly relevant here.