There are two good ways to reach the Lincoln summit from the Quartzville trailhead. The topo map and Photo #1 show both.
LONG ROUTE - Jeep Trail through the Cameron Amphitheater:
This is the easiest hike from the Quartzville trailhead - it is a Class 1 walk-up on mining roads. Leave the trailhead and follow the jeep trail (road) as it climbs west up an easy slope. Follow the road into the Cameron Amphitheatre between Mt. Bross (left) and Mt. Lincoln (right). Cross the Quartzville creek near 12,800' and continue to 13,000' in the Amphitheatre. See Photo #1. Photo #3 shows the road as it ascends the amphitheatre and then up to the shoulder. The road turns right here and climbs northwest up the south side of the East Shoulder. Reach the crest of the shoulder at 13,300' near some mines.
SHORT ROUTE - Directly up the East Shoulder:
This option climbs the entire East Shoulder from Putnam Gulch. From the trailhead, hike west along the road for a bit and then climb directly up the slope that is west of the trailhead. You do not have to stay on the jeep trail to the left. Continue northwest up the hill to 12,100' where you can see Putnam Gulch and much of the route ahead. The terrain on the other side of the gulch is a bit steep but it is still Class 2. Locate a trench (to the northwest) that empties into the gulch - Photo #2. Cross Putnam Gulch and climb up onto the end of the trench at 12,200'. Turn left and begin your ascent of the East Shoulder. Hike all the way up to 13,300' where you will hit the road that comes up from the Cameron Amphitheatre.
From 13,300', continue northwest up the shoulder. There are several mining roads and it's easiest to just stay on the left side instead of following roads to the right across the East Slope. Photo #4 shows the view ahead. Reach the Russia Mine area near 13,800' and the summit hump is dead ahead - Photo #5. Climb directly to the summit or follow the jeep trail north as it crosses Lincoln's East Slope near 13,950'. After crossing the slope, turn left and climb the final 250' to the summit. With snow, this final climb has a difficulty rating of Class 2.