This method worked in the last 2 versions. Haven't tried it in 2006 yet, but I'd assume it will work again.
Here it is step-by-step (a guy seemed confused by my explanation over at Madden Planet):
1. After choosing Advance To The Next Season after your current season has finished, go into the Stadium menu.
2. Choose the Relocate option.
3. Choose the city you want to move to. Choose "Yes" when it asks if you're sure you want to move.
4. Design your stadium, secure funding, and officially submit your proposal to the new city.
5. Now advance to Step 5 of the Relocation Main Menu (the Change Nickname step). Once you have clicked X on Change Nickname, a new screen comes up. The third option down is Team Logo. Choose a generic logo such as the EA Sports logo.
6. Accept your generic logo and advance to Step 6 (Change Uniform) of the Relocation main menu. When modifying your helmet, make sure to choose one that includes a logo (such as Single w/Logo or Triple w/Logo or Short w/Logo or just plain Logo)
7. Once you are completely finished editing your uniforms, hit start to accept your uniform changes and go back out to the Relocation Main Menu. Select Step 5 (Change Nickname). Go down to Team Logo, and select the real NFL Logo or Historic Logo of your choice. Select your logo and move back out to the Change Nickname Menu.
8. Now that you've selected your logo of choice, the game goes back out to the Change Nickname screen with your logo displaying. Open the PS2 tray. There will be a buzzing noise. Close the tray again with the game disk still in the tray. Hit your start button to continue out to the Relocation Main Menu. Go immediately to Step 7 (Finalize). It will congratulate you on your relocation. Sim or play through entire offseason, the next regular season, and the following offseason. Once your team has moved to its new city, it should still have the logo you selected on the helmet.
Note: Logos like Chicago's C appear backwards on one side of the helmet. The Falcon appears correctly on both sides if you use the classic Falcon logo. Those are the only 2 I've tried. Another thing to note is that the logo will be somewhat smaller than usual, but it's not too bad.