Lawyers vary with involvement.
Many are very determined to only sign on at beginning.
My lawyer took over after a nasty JDB violated court order to arbitrate. I was at a loss with what to do. He took over and end ended happily. He was taken aback at my arbitration strategy because he wrote a law journal article pronouncing consumer arbitration dead in my state. After this case he went from declaring consumer arbitration dead, to using it as a strategy because, with my case, he realized how much other side hated arbitration under current, NAF is dead, arbitration condition. (His law firm uses my real last name and officially calls it the [trueq] strategy.)
So this lawyer knows me and my style, so when there is something I need him for, he signs on. He is always fascinated on the case update because he knows I have "unique" set of legal issues.
Much of it is: have you cultivated a relationship with a good lawyer?
That mainly determines when a lawyer is willing to sign on in the middle of a case.