The short answer is no, the people you've described as "future-freed" most certainly are not immune from being taken over by Agents. It's only people who are accessing the Matrix wirelessly (e.g. by broadcasting a hacked signal into the Matrix) that cannot be taken over.
Morpheus makes this abundantly clear in the first Matrix film;
"Sentient programs...can move in and out of any software still hard
wired to their system. That means that anyone we haven't unplugged is
potentially an agent."
The April 1996 Draft of the Matrix script indicates that Trinity certainly considers Neo to be a threat.
She tears off a long strip of black duct-tape and reaches for his face.
NEO : What are you doing?
TRINITY : This has to be done for your protection and ours.
She seals his eyes shut with the tape.
TRINITY : You can't understand right now, but if you're not one of us, you're one of them.
And we also see a possible "future-freed person" named Ash in the Animatrix short 'A Detective Story'. Trinity certainly seems to think that he certainly has the right mental state to have been freed (e.g. without suffering fatal psychological trauma) yet before this can occur, he is overtaken by an Agent.
As Naxa mentions in his comment below, Ash seems to resist being taken over for a few seconds, long enough for Trinity to shoot him. This slight resistance is obviously a long way from actual immunity.
"Trinity: There's a difference between a test and a choice. [She
sighs] For what it's worth, I think you could've handled the truth."