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By Jason Lange

CUAUTITLAN IZCALLI, Mexico, May 26 (Reuters) - Mexico's jobless rate jumped in April close to a 13-year high -- a sign the country's economic downturn has yet to touch bottom as tens of thousands more factory workers lost their jobs.

The number of Mexicans out of work rose to 5.25 percent in April MXUNR=ECI, higher than analysts had expected, the national statistics agency said on Tuesday.

Mexico's economy went into freefall this year as U.S. demand for its exports dried up, leading factories building everything from cars to air conditioners to lay off workers.

"It's a really tough situation around here. I've turned in at least 70 job applications this year," said Sergio Valtierra, 26, as he left an employment office in Cuautitlan Izcalli, a factory town just north of the capital.

An alarming number of companies appear to have closed in Mexico recently, with nearly 11,000 firms disappearing from rolls tracking social security contributions since Mexico began its recession in October.

Mexico sends about 80 percent of its exports to the United States, where consumers have been battered by recession.

The April jobless reading was the second worst since September 1996, when Mexico was clawing its way out of a financial meltdown dubbed the Tequila Crisis. The jobless rate had hit 5.3 percent in February.