From wherever you parked on the road leading toward Modoc Mine, start hiking east and aim for Fillmore Spring. You should get in pretty good on old roads and some trail.
This can be a little tricky, so please bear with me. When I went I could not even identify the summit from below amid the dozens of needle spires. I went with a party that included people who'd done this before. I'll try my best.
On your map find peak 7778 to the southwest of Organ Needle, and identify a very obvious ridge that leads southwest down off of Peak 7778. This ridge 'ends' at Fillmore Spring.
You want to get near this point. If GPS helps, use it. Enough people hike this peak to have beaten in some trails but down low they can be hard to follow sometimes. I'd suggest to generally stay high on the slopes near the rocks and cliffs. A useful landmark is the "Yellow Rocks". Regardless, the land here is mostly open, somewhat thick with cactus and small stands of trees. Cross country is possible. If you get to near the Fillmore Springs, you are in good shape.
If you get lucky you should have picked up a pretty obvious use path near here. If not, start hiking up the ridge SW of Peak 7778, generally staying on the ridge's west side. You very definitely will get the use path by this time. Follow this path steeply up to the saddle shown on the map just northeast of Peak 7778.
From this saddle gaze upon your objective, still 1400 feet above you roughly. The summit is now evident, although the route may not be. Please see the photo with the writing on it for reference.
Here's the deal: one option is to start hiking up these slopes, generally staying left and below the cliffs. There are some paths to follow but this route requires you to crawl through some brush and clamber up many boulder piles and is very tiring. Second option is to stay somewhat centered in this bowl and (hopefully) find a very loose, dusty use path. Either way, this portion is very grueling. Both paths will lead you to a better trail directly beside the cliffs below the summit. Hike up this path, which is in a heavy stand of Maple, and achieve a saddle called Dark Canyon Saddle. This saddle is due south of the summit and is not at all visible from below.
From Dark Canyon Saddle descend the east side about 100 feet, following a reasonable path. A cement slab placed herein a while ago points you to the right access gully. It's only about 1/8 mile of hiking but the dense trees and rocks don't allow for great views. You will generally stay on the cliff/tree margin, wriggle around one large rock, and walk up and into a small pocket. A wall about 20 feet high needs to be upclimbed. It has a good wide diagonal slit and good holds, but is vertical enough to rate a short class 4 segment. Once on top of this section, the summit is an easy 5-minute walk to the west.
On the return our party used rope and belayed some of us down the wall section. It's your personal call.
One way is about 4 miles with about 4,000 feet of gain.
Good boots, long pants. A rope and harness might be useful to belay over the wall. Bring enough webbing in case you want to set a simple anchor. Many people just free climb this section.
If you have information about this route that doesn't pertain to any of the other sections, please add it here.