Yes. I could make this work! Here's what I did.
Oh, be sure to keep the nails and brads that you pull out of the basket. You can use these later when assembling the hat. I, personally, only used the brads, but by all means, use the nails if you can.
Once I had taken the basket apart, I had a pile of balsa wood strips. They're looking a bit silly right now, but we'll fix that soon.
Here you can see the pattern that I made. It's basically a large circle with a slice cut out, forming something of a "dart". I wish I could tell you the dimensions I used on my pattern, but I made this a while ago and didn't take notes as I went. :-/
In the picture above, you can see something of a frame taking shape. This was made using the top edge of the basket, which was the thickest part of the Easter basket. It was double layered, so I got two long, sturdy strips out of it.
Be sure only to make your frame up to the edges of your dart. You can see above that my frame is a little shorter than the pattern I made, but it's not a big deal. This isn't an exact science.
This is when the brads came in useful. Since the basket handles were attached to the basket at the ends of the top edge strips, they already had holes drilled in them to keep the handles secured to the basket, and I could match the holes in the strips and use the brads to secure them to each other. It's like the basket wanted to be a hat all along.
Moving right along, once I had the pattern figured out, it was time to prep my balsa wood. I set a large pot of water on the stove and heated it up to boiling, then took it off the burner and let it cool for a few minutes. I wanted the water to be hot, but not scorching, so I wouldn't hurt myself while softening the wood.
Once the water was at a workable temperature, I could set strips of the basket into the pot and let the heat and moisture soften the wood. This took the kinks out of the strips and left the wood pliable.
Here you can see the before and after for one strip of balsa. On the left is the strip before being soaked in the hot water - very gnarly and wavy. One the right it's lovely and straight. It still has a bit of waviness to it, but it looks much better than what we had. It's also soft after being soaked, so it will be easy to work with.
The first part of building the hat was to get my spokes in place. I used the longest strips of the basket to do this. They don't all have to be the same length, don't worry about them all being even. As long as they are generally the same length, you'll be fine. If some of your strips are really too short to reach all the way from one side to the other, snip it so it will at least go to the centre. (What you cut off can be woven into the hat later on.) As you add each strip, secure the overlap at the centre with a dab of hot glue. I did this each time I added a strip so the centre would be sure to hold its shape. I didn't want there to be a chance that the hat would fall apart.
See? Random. You can cut up your longer strips to fit into spots that seem to be lacking. The wood will be soft enough after soaking it that you can do this with a regular pair of scissors.
Once you have a bit of your hat filled in, go ahead and close your "dart", by bringing the two sides of your opening together.
Go ahead and dab some glue on the bits that overlap to secure the shape of your hat.
At this point, the hat is looking a little sparse. Weave some more of your strips into the hat, moving around the hat as you go. Be mindful that you don't put all your strips going in the same direction - go diagonal in both directions, vertical, horizontal, back to front, over the top, etc. Just don't be messy.
Try to cover up the join in the back where you brought the two sides of your "dart" together. This will help disguise it, and add more stability. You can also weave strips over the top of your hat, to help cover up the ends of some of those shorter strips that ended at the centre.
Here you can see the top of the hat, where I've started to weave additional strips over the centre join to hide the bit of messiness left by the shorter strips.
I tried to use as much of the strips from the basket as possible when weaving my hat. I didn't worry about the strips ending abruptly, just making sure that the ends of any short strips were tucked behind other strips to disguise them.
As you can see, I have some pretty funky ends going on. Not to worry, these will be trimmed when I add the bottom frame.
Now it's time to join the hat to the frame. Overlap the bottom edges of your hat on the frame and start gluing the bottom edges of your strips to the frame. Don't worry if some pieces hang farther over the frame than others. Just make sure that all your spokes are glued glued onto the frame.
In the pic below, some strips have been glued down to the frame. The strips on the very right of the picture are not yet attached.
Once you have attached the entire hat to the frame, trim off the excess that's hanging down past the bottom edge of the frame.
Congrats, your hat is pretty much finished!
To finish off the hat properly, I braided some long grasses and glued them onto the bottom frame to hide the edges of the hat that had been glued down. I just picked some long grass and used that, but there are long decorative grasses in the floral departments of craft stores that would work just as well.