The days when airlines offered 'bucket' fares to last-minute travellers are long gone. The budget airlines require you to book well in advance to get the cheapest fares.
The cheapest flights from East Midlands Airport might well be with Ryanair, to Dublin. That's nominally a 1 hour flight but you'd actually be in the air for no more than 45 minutes.
As an example, you could take a day trip on Monday 19th September for a total of £33.98 (return, including taxes & charges) plus their debit/credit card booking fees of 2 x £6, making £45.98 in total. (You'd need to add the bus fare into the city centre on top of that, but it's still a cheap, and interesting, day out). You'd need to get up early to catch the 0635 flight outbound. The return flight takes off at 2215.
I would have investigated some alternative dates but Ryanair's booking system has just gone down. Instead, here's a list of the closest destinations served from East Midlands:
Belfast (1hour flight, served by BmiBaby and FlyBe. Day trips are possible with both airlines but BmiBaby has better-timed flights if you're only going for a day).
Glasgow (1 hour 5 mins. BmiBaby. Day trips possible)
Edinburgh (1 hour 10 mins. BmiBaby. Day trips possible)
Amsterdam (1 hour 25 mins. BmiBaby. Day trips possible)
Brussels (1 hour 30 mins. Bmi. Day trips possible)
Jersey is 1 hour 20 mins from East Midlands, Guernsey is 1 hour 30 mins but you're unlikely to get cheap flights. Newquay is 1 hour 15 mins away but the flight times wouldn't permit a day trip. Dinard (France) is only 1 hour 10 mins away but you couldn't do a day trip. Paris Charles de Gaulle is 1 hour 30 mins away (with day trips possible through BmiBaby) but you might find tickets are expensive.
http://www.eastmidlan.../content/Destinations
Chris