If you do not re-sign the player by the end of this season, you may or may not be able to sign the player before free agency begins the next season. If you have taken a player to arbitration 3 times, the player is most likely going to test free agency. If it is a player who you signed as a free agent who is up for another contract, you should have an opportunity to re-sign the player -- depending on the success of your team. If it is a really good free agent and your team is struggling, he might test the waters. If you are contending, he might give you a shot. You can only trade a player under contract, so you would need to do it this season.
If it is a player you really want to re-sign, do it now.