I don’t think he’s far enough left to consider switching parties. Besides, what would be the benefit for him even if he did it purely for political reasons? He represents a heavily Republican state and he would be the odd man out in a new party.
Besides the party switching tide in recent history (1975 to present )is heavily in the other direction. The following is from Wikipedia
Democrat to Republican
1975 - John Jarman, while U.S. Representative from Oklahoma. He had served for 24 years in the House and said he was fed up with the Democratic party, which had been “taken over by liberals”. He retired in 1976.
1979 - Jim Gerlach, During the Carter Adminastration. He is a United States Rep. from Pennsylvania
1981 - Bob Stump, while U.S. Representative from Arizona
1981 - Eugene Atkinson, while U.S. Representative from Pennsylvania
1983 - Bob Martinez, while mayor of Tampa, Florida
1983 - Phil Gramm, while U.S. Representative from Texas (resigned before switching parties and re-won his seat in a special election)
1984 - Andy Ireland, while U.S. Representative from Florida
1985 - Kent Hance, after losing the Democratic U.S. Senate nomination in Texas
1988 - Jim McCrery, while running for U.S. Representative from Louisiana. He had been a staff member for Buddy Roemer, but switched parties before the special election after Roemer was elected governor.
1989 - Bill Grant, while U.S. Representative from Florida
1989 - Tommy Robinson, while U.S. Representative from Arkansas
1990 - Rick Perry, while running for Agriculture Commissioner of Texas
1991 - Buddy Roemer, governor of Louisiana switched parties shortly before the beginning of his reelection campaign, which was not successful.
1994 - Walter Jones, while running for U.S. Representative from North Carolina
1994 - Mike Bowers, while Attorney General of Georgia
1994 - Fob James, while running for Governor of Alabama
1994 - Dick Shelby, while U.S. Senator from Alabama
1995 - Jimmy Hayes, while U.S. Representative from Louisiana
1995 - Greg Laughlin, while U.S. Representative from Texas
1995 - Ben Nighthorse Campbell, while U.S. Senator from Colorado
1995 - Billy Tauzin, while U.S. Representative from Louisiana
1995 - Nathan Deal, while U.S. Representative from Georgia
1995 - Mike Parker, while U.S. Representative from Mississippi
1996 - Norm Coleman, while mayor of St. Paul, Minnesota
1998 - Sonny Perdue, while a Georgia State Senator
2000 - Matthew G. Martinez, while U.S. Representative from California
2001 - Michael Bloomberg, before running for mayor of New York City
2002 - Amy Tuck, while Lieutenant Governor of Mississippi
2004 - Ralph Hall, while U.S. Representative from Texas
2004 - Rodney Alexander, while U.S. Representative from Louisiana (his switch just before the filing deadline prevented the Democrats from fielding a viable replacement candidate)
Though he never formally changed his affiliation, former U.S. Senator Zell B. Miller (D-Georgia), caucused with the Senate Republicans and spoke at the 2004 Republican National Convention. Former Democratic New York City mayor Ed Koch also announced his support for Bush.
Republican to Democrat
1977 - Peter Peyser, after three terms in the House as a New York Republican, vacated his seat to run for nomination for the Senate in 1976. After his unsuccessful attempt, he switched to the Democratic party and regained his House seat in 1978.
1991 - Mike Doyle, now U.S. Representative from Pennsylvania
1992 - Loretta Sanchez, now U.S. Representative from California
1996 - Carolyn McCarthy, to challenge Dan Frisa (she was still a registered Republican at the time, but changed her registration to Democratic in 2002)
1996 - Russell Peterson, former Governor of Delaware
1999 - Michael Forbes, while U.S. representative from New York
2002 - Ray Nagin, days before filing for the mayoral race in New Orleans
2003 - Teresa Heinz Kerry, in protest of the campaign tactics used by Saxby Chambliss in the Georgia U.S. Senate race of 2002
2003 - Barbara Hafer, while State Treasurer of Pennsylvania
2005 - Andy Warren, former Bucks County, Pennsylvania Commissioner and well-known local politician (now running for Congress)
2005 - Paul Morrison, Overland Park, Kansas Attorney; now running for State Attorney General
The numbers work out to 31 switches from Democrat to Republican and 11 from Republican to Democrat since 1975. The difference is even more stark when you focus just on the US House and Senate. In that case the difference is 22 to 6.