I think this is incorrect.
I believe if you signed said player to a 1-year deal (the only option) from the Free Agent Pool, then you cannot offer them an extension until after that season is over (and yes, in this particular case, you only get one chance, no back-and-forth negotiating).
Now, on the other hand, if you acquired a player in the final year of his deal from another team, you may or may not get the option to resign them to an extension during that final season of their contract. I'm not sure if this is a glitch or intentional, I think it's allowed me sometimes, but not allowed me at other times. Uncertain on this one.
Lastly, if you signed a player to a one-year extension during the previous season (so you're basically constantly extending his contract by 1-year every season), I believe you should have the option to offer them an extension midseason. I personally did this with Brandon Browner for several seasons to keep his contract numbers reasonable (boatload of guaranteed money + the coaching perks), without getting locked in long-term if his stats declined heavily any given season. Likewise, with all the glitches Madden 25 has, I'm not sure the game would credit you back the guaranteed money if you signed a player to say, a 5-year contract with $25-40 million guaranteed, and then he retired the following season. That guaranteed money might just up and vanish in a puff of smoke. I don't know this for certain, but I think you can tell that I don't trust Madden's programming to properly account for this.