A free agent can lower his "asking price" whenever; and unless I'm very sadly mistaken, a player never stops being a free agent unless they've signed a contract.
Sometimes they'll eventually get a job; a lot of players will sign after the season's already started.
The vast majority of these players are those who get minor league contracts and aren't going to be making the majors, and a lot of times it's because the organization they were playing for released them outright; Gil Velazquez was released by the Yankees May 10 last year, and the Marlins signed him May 18. He's pretty typical, playing 101 games in the minors in 2013, and appeared in "one" MLB game.
There are the occasional free agents who just choose not to sign until mid-season, but they're older, already established and the kind of player teams could actually use during a pennant race. Roy Oswalt has waited to sign until May in both 2012 and 2013, and Roger Clemens did that the last two years before he retired (for real).
A lot of players will sign minor league contracts, some of whom are invited to spring training as a non-roster invitee, which basically means the team is letting them try out for a spot on the MLB team but don't want to put them on the official roster (those spots are important).
If a player can't get a contract with an MLB team or it's minor league affiliates, they might go to a foreign league, like Japan or Mexico, or independent leagues; the Long Island Ducks of the Atlantic League (all teams are not affiliated with MLB whatsoever) has become well known for signing former MLB players.
But there are always players who will only sign for a certain amount and thus don't get signed, or won't sign because they won't take a minor league contract, or (and when it's a player with a long career who I like, it's kind of sad) just never get an offer at all.