How long does your current linear method take to get from stopped to full speed and what is the preset constant?

the closer it is to full speed the shorter the time between steps is

I'm not sure that this is what you should do - it probably won't look right. I think the acceleration at the beginning and end should be lower than it is in the middle. But you'd have to experiment with it to find the right acceleration curve.

One way to approximate the exponential curve would be to break it into several linear pieces. At the moment, at any point in the acceleration (and, presumably, deceleration) you are using the same linear constant to increase the speed. You could keep track of what stage of the acceleration you are at (as you do now) and map that into one of, say, five constants such as 1, 3, 7, 3 and 1, or whatever is appropriate. If it doesn't take long to get up to full speed I doubt that you'll notice the difference between that and a true exponential.

Pete