The snap tool is a good tool but it does not work for everything. If you can’t make it to work you have to find another way.
To make the shrink-wrap (and also the mirror) visible in edit mode you have to click the vertex button In the modifier tap. (red circles)
Delete all the vertices where the green line is. (‘x’ > dissolve vertices ) so you have only one vertex left at the top. (Red dot) Select that one vertex at the top, press ‘g’ and slide it to the position you want. It will snap to the surface of the sphere because of the shrink-wrap modifier.
If you have this top vertex where you want it, select that edge (green line) and subdivide it a few times to have more vertices again.
You can enable/disable every modifier in ‘view’ mode by clicking the ‘eye’ button’. The camera disables/enables the modifier when rendering. (Green arrows)
The sphere does not have to be on the same layer. You can move it to another layer and disable that one.
Edit, for the mirror to work propper you have to move it the top of the stack. Use the up/down arrows on the right of the modifier.(Sorry 'bout this)