Considering how long security can be at MDW, I'd be wary of having someone with only 40 minutes free leave the secure area and have to clear again.

You might be able to sweet talk a ticket agent into giving you gate passes, but if that fails then you've gone to Midway for nothing.

As nba1017 mentioned, buying refundable tickets, using them to clear security, and then getting them refunded seems to be a popular option, although I've never tried it. It will tie up some $$ on your credit card until the refund clears, since refundable tickets tend to be expensive.