Chavez does well versus right-handed pitchers. He his .299 with a .366 OBP and a .545 slugging. Against lefties he did not do as well .152 AVG, .231 OBP, and a .152 last year. For his career .275, .356 OBP, and a .501 SLG. He needs a platoon partner and my suggestion is Jeff Keppinger.
Keppinger is a free-agent, which is great for the Yankees since they don't have many trade chips. He also should come at a relatively cheap price, 2012 salary was 1.52 million, while posting a 2.8 WAR. His slash line last year was .376 / .402 / .521 vs left-handed pitchers. .302 / .352 / .403 vs righties. For his career he has a slash line of .333 / .376 / .487 against lefties.
Keppinger also provides some position flexibility with his ability to play short and second base.