You want to upgrade your axe, not your pickaxe. You can chop the big stumps after the first upgrade and the big logs after the third. For starters, you can get a fair amount from the big stumps and fallen logs on your farm. The Secret Woods has six big stumps that reappear daily.
As soon as you can get seeds from trees, you have a chance at getting mahogany seeds from all those stumps and logs. Plant them anywhere a wild tree can grow to get mahogany trees, which also drop hardwood (I always plant a few around the pond in the Secret Woods since I go there when I need hardwood anyway). If you have the recipe for tree fertilizer (or got some by any other means), use it on them.
If you choose the Forrester profession at Foraging level 5, all trees, logs, and stumps will drop more wood. At level 10, the Lumberjack profession makes all trees have a chance to drop hardwood alongside regular wood.
If you catch a woodskip in the pond at the Seacret Woods, you can drop it in a fishpond for a chance at it giving you hardwood. Hardwood can also be found in fishing chests or in brocken crates/boxes in the mines. If you have a dog on the farm, it has a chance at gifting you hardwood after reaching five hearts with it.
If you don't mind cheating a bit, you can access the Secret Woods before the third axe upgrade by placing a chair directly beside the bottom of the fallen log blocking the path, sitting in it, and hopping out on the other side. Make sure to use the cursor to retrieve the chair so you can use it to get back out, otherwise, you'll be stuck in the Secret Woods until you pass out at two AM.
