The reason you are having trouble changing the color of the hat is because there is very, very little texture or any other detail (eg, shine, side-to-side brightness variations) in the hat to start with. It is almost perfectly black, so that if one tries to change it into a bright color, you are going to magnify all of the sensor noise as well as jpg and other artifacts, so it's just not going to look like cloth any more. This may seem to be a simple challenge, but since you are starting with almost no data in that area, it's more difficult than you may think.
If you want it bright (and realistic) your best bet is to find a photo of cloth (or even a hat) with the texture, color and brightness that you want, and then insert it into your image using the same techniques one would use to reconstruct a old damaged photo with missing areas. This can be complicated, but fortunately there are many tutorials and threads on this, so I'm not going to elaborate further here.
However, if you would be happy with darker colors, you can probably get away with straightforward moderate brightening of the current, almost black hat followed by weak colorization. One can try to utilize blending modes (as suggested above), but I took a more straightforward approach: a couple of "levels" adjustment layers (with blendIF sliders used to control contrast, ie, control areas which might become overly bright), followed by one hue/saturation adjustment layer set to normal mode, and a second set to "colorize" mode to introduce color where it really doesn't exist. I also used blendIF adjustments on the hue/sat layers to control the application of color to certain tonal ranges. The steps in the process, as well as a couple of different color options are shown in the attached GIF. It's far from perfect, but since the remainder of the image is a bit OOF and has other problems, you might be able to get away with this approach.
HTH,
Tom M
PS - Since I didn't know how much texture that particular hat really has (depending on the part of the world it is from), I guessed and gave it quite a bit of texture - almost wool-like. If it actually is smoother, then one would use slightly different adjustments -- less contrast in the hat after brightening, more emphasis on the introduction of untexturized color. If you could provide a good quality photo of the texture/shine/brightness/color you are looking for, we would have something definite to work from.