Alexander, a second-round pick of the Detroit Lions in 2007, played two games with the Miami Dolphins earlier this season.
He also has played for the Jacksonville Jaguars and Carolina Panthers, compiling 30 career starts, four interceptions and 159 tackles.
The Jets had to make a move because Leonhard's injury left them perilously thin at safety. In Sunday's win over the Kansas City Chiefs, he tore the patellar tendon in his right knee on a second-quarter interception. Leonhard will have surgery Wednesday morning.
This marks the second straight year that Leonhard will finish the season on injured reserve. He broke his right leg last December in a freakish practice collision and missed the final five regular-season games, plus three playoff games.
"It's obviously frustrating," Leonhard said Tuesday morning on WFAN radio. "It's not the way you envision your season ending. It's the risk you take when you play this game. It is frustrating, especially the second year in a row -- and you have no reasons why."
Rich Cimini covers the Jets for ESPNNewYork.com.