I don't believe you can take the lift up from the first level (58 Tour Eiffel) to the second level. And you definitely have to pay additional for tickets from the second to the top.
We had reservations at 58, arrived early, so we took the lift up to the second level. I took the lift down to 58 to see if the restaurant had opened yet, leaving my wife and grand daughters on the second level. To my dismay, I found that I could not take the lift back up to the second level, but could get there only by walking up the stairs. I remember this distinctly because my fear of heights made the open stairs pretty scary. I would definitely not have taken the stairs if I could have used the lift.
That was 4 years ago, so things could have changed, but that's how it was when we were there. For what it's worth, the stairs aren't all that scary for everybody. You might enjoy the adventure. Lots of people were enjoying them, in both directions, while I was trying not to fall to my death.