A printer does not really have a standalone gamut. In practice, gamut is the result of the printer/ink/paper combination, so the useful gamut is always tied to a specific paper profile.

That means the ICC profiles you’re checking are the right place to look. If you want the most accurate answer, print a profiling target on the paper you care about and measure it with a spectrometer/colorimeter to build or verify a custom ICC profile. Published ICC profiles for common paper/ink combinations are the next best option.

For improving vivid-color prints, glossy or luster papers usually offer a larger gamut than matte papers. Epson Ultra Premium Photo Paper Luster was specifically mentioned as being among the better options.

Also, if many colors are out of gamut, try Perceptual rendering intent instead of Relative Colorimetric when printing or soft-proofing. Perceptual compresses colors to fit the printable gamut more smoothly, which can help with highly saturated flowers.

You can inspect ICC gamut volumes in profiling/viewing software, but remember those gamut tags are not always perfectly accurate.