I got a walk up permit two years ago for two people the morning we left. (We came back and day hiked the section down to the valley after Whitney.)
My suggestion if it fits with your plan. The park releases the non-reserved permits the day BEFORE the departure day based on trailhead quotas. So if you can get there the day before you can increase your chances. Also, the fewer in the group the better, solo hikers have it the easiest. Since the quota is by trailhead you have several different trailheads to choose from, not just Lyell Canyon. If one is taken then move on to the next. Finally, if you have flexibility then pick a mid-week day.
I always due walk up permits at Yosemite and have been able to land every trip I have wanted. Good Luck
PS – You are better off getting there the day before than early in the morning. They don't open until I believe 7am and many are already filled from the day before. Then at 10am all the slots are released from the reservations not picked up. That is the second good window.