By Kristina Cooke and Nancy Leinfuss

NEW YORK, May 5 (Reuters) - The Federal Reserve's premier program to revive securitization markets and spur consumer and small business lending showed signs of life in its third round after an anemic start in March and April.

The program is a crucial part of the Fed's efforts to lower borrowing costs for consumers for everything from student loans to a new car, which had sky-rocketed after credit markets froze last year.

Investors on Tuesday requested $10.6 billion in loans in the May round of the Term Asset-Backed Securities Loan Facility (TALF). It was the strongest showing yet and is up 66 percent from the first two rounds combined, but still a small portion of the $200 billion the Fed has pledged to lend in the initial phase of the program.

"It's still a fraction of what they said they could lend ... but it seems like it finally got off the ground this month," after teething problems, said Stephen Stanley, economist at RBS Greenwich Capital. The Fed has said the TALF could grow to $1 trillion.

Through the TALF, the Fed makes loans to investors for the purchase of asset-backed securities. In the run-up to the May round, companies issued almost $14 billion of TALF-eligible securities backed by credit card, auto, motorcycle, equipment and small business loans -- more than twice the amount of the past two rounds combined.

Concerns that the government could retroactively change the terms of the deal if investors are seen as profiting too handsomely from it combined with the complexity of the paperwork slowed the program out of the starting block.

Analysts said they were encouraged by the jump in demand in May compared with April. Investors requested just $1.7 billion in April and $4.7 billion in March.

Analysts said interest would likely further increase once the Fed includes commercial mortgage backed securities in the program in June. The Fed has also said the program could be expanded to include lower rated, older assets as part of a comprehensive government plan to remove so-called toxic assets from banks balance sheets.