Note: We're working with two assumptions here - the first being that you don't have a high clearance 4WD available and the second being that driving to the Carbonero Mine is no longer allowed but walking in is. Total mileage and elevation gain reflects these assumptions.
From the trailhead coordinates provided, try to locate and follow the old road up to the Carbonero Mine. If able to drive, park at a flat area near the mine at about 11,450 ft. From the mine, the old road does continue up the ridge a little further, but at some point, you might as well abandon the road and hike directly up the steep ridge and follow it all the way to Pt.13,446. Above the mine, hiking continues at first on grass with a few trees, then tundra that eventually gives way to scree and rocks. A little below the point, there are some minor cliff bands to work through and things steepen up even more. At Pt.13,446, you might assume you've arrived at the San Joaquin summit until you look east. About 1/3 mile away, there's an obviously higher "true" summit with an extrapolated elevation.
Hike over to the true summit along the ridge crest, losing about 250 vertical feet in the process that you'll have to regain. There will be more rocks to walk over and another minor cliff band to break through near the summit. Admire the rugged and dramatic "San Joaquin Ridge" that protrudes north into the headwaters of Bear Creek. The lower portion of the Bear Creek drainage is ringed by prominent cliffs. The upper basin is a tundra paradise ringed by rocky peaks and ridges. If this is your only peak for the day, either return by the same route or drop directly south from the summit and then work your way back over to the Carbonero Mine and continue on down to the start. Otherwise, see the directions for Silver Mountain which lies west along the ridge and add an interesting challenge to your day.