You can redeem this, provided you have access to wood or building materials, in a simple and elegant manner that shouldn't take too long.
You can use construction materials (wood or stone or glass or soap or metal) to construct staircases (rather than dig them).
Constructions - specifically up, down, and up/down staircases as needed. Make sure you have a dwarf with the relevant labour enabled (carpentry for wood, smithing for metal, or masonry for anything else). Dwarves are able to stand on a downward staircase and construct an up or up/down staircase in the empty space below.
Once there's a route down to your dead dwarf, you'll be able to get to his pick, and a new dwarf can start mining. You may have to unforbid the pick, and his other belongings, as it may have been automatically forbidden upon death. If you're feeling really nice, you can entomb him.
If you need construction material, deconstruct workshops as needed. If you still have your starting wagon, deconstruct it for 3 logs. If you have an axe, you can have a woodcutter gather you some fresh tree.
I Have no Axe
First of all, never admit this. Secondly, we can fix this, as long as you have 1 log and 1 building material (which can be another log). If you've still got your wagon, it's got everything you need.
Build a carpentry workshop and make a wooden training axe from the log. It's not an impressive weapon, but a dwarf will still be able to fell a tree with it. There are trees nearby, right?
If you haven't got your pick back yet, I don't know how to help. Good luck!