When I saw this thread, I thought it was my old thread bumped... I asked the same question.
I use, I guess, the 'ball' of my thumb and the side of the tip of my index finger, sometimes I pinch the contact partially through my eyelids. Squinting at the right time can help some too. It's just something you'll get used to as time goes on, it's quite possible. Just don't use the nail, in my experience it never works.
One time I had a lens stuck to my eyeball, I could not get it out. So I slept with it in and the lens became lubricated enough over night to come out effortlessly the next morning. Contacts are an inexact science.