The only normal image format that's supported by MATLAB for writing float data is TIFF.
inpict = imread('peppers.png');
inpict = im2double(inpict);
filename = 'testtifffp.tiff';
tagstruct.ImageLength = sz(1);
tagstruct.ImageWidth = sz(2);
tagstruct.Compression = Tiff.Compression.None;
tagstruct.SampleFormat = Tiff.SampleFormat.IEEEFP;
tagstruct.Photometric = Tiff.Photometric.MinIsBlack;
tagstruct.Photometric = Tiff.Photometric.RGB;
if strcmp(class(inpict),'single')
tagstruct.BitsPerSample = 32;
elseif strcmp(class(inpict),'double')
tagstruct.BitsPerSample = 64;
tagstruct.SamplesPerPixel = sz(3);
tagstruct.PlanarConfiguration = Tiff.PlanarConfiguration.Chunky;
newpict = imread(filename);
Don't expect any other application to be able to read it though. Given that float TIFFs aren't widely supported, you might as well just save it to a .mat file if you really needed to save it in floating point.
Also, if you're just trying to scale a uint8 image to unit-scale (0-1), then using rescale() is probably the wrong thing to do. When used in the given manner, rescale() does not map [0 255] to [0 1]. It maps whatever the image extrema are to [0 1], which will usually change the image contrast. Tools like im2double() and im2single() are what you use to change numeric class and scale without altering the image data relative to black and white.
That said, it's not clear why you can't store the image as uint8. It's already been quantized, so storing it in unit-scale doesn't accomplish anything but increase the file size and make it incompatible with most other software. In the given example with peppers.png, the TIFF file is over 16x as large, but it contains no more color information than the original PNG.