Fairy fountains will only spawn fairies when the amount of fairies in your inventory is at 2 or less, but when they appear they will not disappear unless you scare them away. This means that if you hold enough fairies in your hands to make the inventory counter go down to 2 then more will spawn in a few seconds, then you just need to put them back in your inventory and collect the others near the fountain. If you drop 1 fairy and hold 5 in your hands so your inventory counter is at 2, after waiting a few seconds 4 more will appear that you can get after you put the fairies in your hands away and rush up to catch the stray one. This will give you 12 instead of 6 as the maximum amount of fairies.