Hi,
Resizing a hat can often be a tricky business, even for a professional milliner. Even if you successfully stretch the crown to fit her head, it may warp the brim which is hard to get "right" again. Also, the crown is narrower at the top than the bottom, and so stretching it by hand (without a block) may also warp the sides of the crown. Your hat is a true Panama, and so check to see if it was blocked in 2 pieces, with the brim and crown blocked separately and then the crown and brim sewn together. If it was, then stretching it more than a tiny bit will probably break or pop the connecting stitching and create a much larger problem to fix. Often a hat can still be stretched without that happening, but it should be done on an electric hat stretcher which heats up and applies equal pressure all around at the same time, with heat. If you did it on a stretcher then the ribbon may not need to be removed. There are still a handful of hat shops in California that still offer this service. I will think of the names. Or you could try steaming it yourself (be careful of burns) and then placing it immediately over a hat block and applying gentle even pressure with your hands, smoothing it to stretch the straw fibers. Repeat the process until the hat is stretched enough (or the threads pop!

). Let the hat cool on the block. OK, now you ask...where do I get a hatblock in her size? If she has a truly big head as you say, this is not so easy. Larger size hatblocks are usually for men and are in great demand and very expensive. So, you could use a substitute such as a large metal bowl or pot, marking her head circumference on the bowl somehow. This should work. Some use aluminum foil to cover the pot or bowl, but I have not tried this. Panama is a very strong straw but so thin and light colored so make sure there is no chemical reaction that may darken the straw's color. Do not put a damp straw hat in a microwave to steam it, as some people might suggest to you! It will cook the straw and harden it. That can work for some types of fur- felt hats, but I do not recommend it at all.
Does the hat have a fedora crown or is it flat on top? A fedora can be trickier than a round or flat top.
If it were my hat I would separate the crown from the brim by picking out the stitches, and stretch each separately. Remove the ribbon of course. They have to be the EXACT same size or when you sew them back together or...well they won't fit! Of course, even I would be hesitant to attempt this feat. It is not a favorite thing to do. And I do not have the right size hat blocks, my largest are size 23. They do make sleeves you can put over the increase the size, but now we are getting into another area and more expense.
If you stuff it while steaming, use paper, not plastic as that steam is very hot.
I have never heard of using the ribbon to resize a hat, I am not sure what that means. The best case scenario would be if you could, as Hollis suggested, take out the headsize ribbon and try to gently stretch the hat by hand. If only a few of the stitches pop, it probably would not harm the hat. You could also make a small cut in the stitching in the back of the hat, but this raw cut may get larger over time and is not the best way. I have done that on a few inexpensive hats as it is the fastest and laziest way.
If you replace the ribbon, make sure it is millinery grosgrain ribbon as regular grosgrain won't work and it does not stretch or swirl to the shape of the crown. But you probably know this from your hat making days!
May I ask, what is her headsize?