Saddle soap is great stuff but it's been so long since I've seen it that I don't even look for it anymore, not in this city.
Mink oil is easy to find; it comes in a shoe-polish type of can.
I would brush that hat real good with a soft natural-bristle brush, rub in some mink oil with a soft cloth (old tee shirt or similar) folding and refolding as it picks up the dirt
with particular attention to the seams. Let it sit for a few minutes, then brush again to polish the surface. Wait a day or two and see how it looks. You can reapply the mink oil in thin coats until you get a soft even gleam.
edit: you are dealing with smooth leather, yes? Because suede needs a stiffer brush and a different product.