Second pass lith works if it has the look and colour you want. Or if the paper you like doesn't lith properly.
That said, I have some lithable papers that DON'T lith in a second pass lith (that is, they look exactly the same before and after), despite being very responsive to lith in a first-pass. I've also read of papers that looks the same first- or second-pass lith (but they may be discontinued now). If so, for me, what's the point of second-pass lith? It's much more streamlined to just lith the print straight up instead of having to go through all the extra steps of making a print, (wash and/or dry) bleaching back, (wash and/or dry), redevelop it (wash and/or dry), and then possibly tone it.