Do hospitals use blood as soon as they receive it?

I work for a courier company who uses Independent Contractors as drivers. We have an account with a customer that supplies hospitals with blood. At times they say they need a delivery super stat because there is someone "on the table". I suspect this is a ploy to try to expedite the service. I refuse to believe that hospitals begin any surgery without having blood on hand. When we receive the blood it's packed with ice and boxed up pretty tight. Am I right in believing that hospitals are not relying on a third party non-emergency courier service for critical "on the table" surgeries?