I don't think that's true. I'm almost certain that partial no trade lists have to be submitted in the off-season and are locked in until the following off-season when they can be changed.
I can almost clearly remember headlines saying, "player X has been asked to submit his X team no trade list." or something along those lines, mid season.
I could be wrong and am not willing to do the leg work, but take my word for it, I might remember it happening....
Edit: I lied, I did the leg work
"Some no-trade clauses are modified, and the types of modified no-trade clauses vary from player to player. Some players must submit a pre-determined number of teams they will accept a trade to if they are asked. In other modified no-trade clauses, players must submit a pre-determined number of teams they will not accept a trade to if they are asked.
The lists of teams to which a player will or will not accept a trade do not exist ahead of time. The lists of teams themselves are not a part of a player's contract. A player creates the list when he is asked to by management.
In no-trade and modified no-trade clauses, if the player does not submit a list of teams in the specified time frame after being asked, he can be traded anywhere."