Most airlines offer "anytime" awards for more miles than the "saver" awards. ANytime awards are essentially Y class bookings in coach - if there are any seats available, you can get them - and are usually about the same # of miles as the saver awards in 1st. So if this is really important, just ask for anytime awards if the savers aren't available.

In addition, on UA, an anytime award gets you a seat in Economy plus section, even if you are not Premier.