Our experience was that fastpass availability during EE was hit and miss. Some were open as soon as the park opened, some 1/2 hr or so into EE and some not at all.
Since EE is only an hour, every minute counts. We have been known to get one when the return time was only 10 or 15 minutes away and when the regular line no longer than that. Since you have an hour window to return, you have plenty of time to go do other things before using the fastpass. We found that the lines lengthened pretty quickly during EE, so it was impossible to do more than 1 or 2 popular attractions without having to stand in line. Getting a fastpass as soon as possible during EE also allows you to get another fastpass very soon after the park opens to the general public.