If a dwarf is special to you because of some specific high skill level or special history with you, treat them special, as they're an individual.
If those 'unorganized dwarves' are essentially identical to you (and in the work they can do), what I do is turn on almost all labors for the lot of them. Since they can't be miners and woodcutters both, I generally select for a specific number of miners and all others are assigned woodcutting (or neither labor if they're meant for the military). Hunting and fishing are turned off for all dwarves as a rule, to ensure I know they're very unlikely to randomly wander out of my controlled areas. Healthcare is turned on for only those with the highest skill in that labor in the fort (no one knows surgery? No one has that labor yet then. I had someone with a diagnosis 3 and a migrant comes with diagnosis 4? Only the best for these jobs).
I control laborby one of two means. If I want a workshop-handled task done only by a single highly skilled dwarf, I make sure I have a manager on the nobles screen and use the P option on the workshop to specify either that dwarf, or that dwarf's skill level. If I want a non-workshop handled task done only by the best, then I use dwarf therapist to turn that labor off for all the rest, at least during that time.
The advantage of this is that for 'general labors' (build these things, haul those things, go build 300 segments of wall or whoa, I need 100 cage traps placed pronto), I have an entire pool of labor, almost my entire fortress, ready to do it. In general, I want the work done now, not done very slowly but very well. And the (for me very few) other things that I do want done with a specific level of very high skill, however fast or slow that goes, are specifically and carefully limited.
Edited to add - just realized that you might actually be asking 'what to do with all the idle hands, what should they be doing?'
For that, I'd urge you to slowly explore whatever workshops and designations that you've not really explored yet.