Chrysler began shutting its 30 manufacturing plants on May 1, a day after filing for bankruptcy.
Economists estimate that Chrysler has laid off 27,000 workers since then.
The firm is to be revamped after it agreed a deal with the Italian vehicle manufacturer, Fiat.
The number of people staying on the benefit rolls after collecting an initial week of aid jumped 202,000 to a record 6.56 million in the week ended May 2, the Labor Department said.
That was the largest weekly increase since late November and the 15th straight week that "continued" claims reached a record high.