If you are injecting daily your best bet would be to do as Mark suggested.
Even when using a 31G drawing should still be quick as the volume of oily solution will be minimal when injecting daily.
Takes next to nothing to draw 5-15 IU.....even then how many on dailies are injecting 15 IU (210 mg T/week) not many!
Hard to believe anyone would complain about drawing such a low volume.
I could see if one was drawing 50-100 IU then it would be much slower when using 31G.