I can imagine a very special red color in my mind but I can't find its color code in Photoshop or any other online color dictionary. Is there any tool that for example I say: "Hey show me all possible red colors exist in nature slowly so I can compare result with my taste to choose one"?
Short answer: no
Long answer: "Red" is very subjective:
What you're thinking of "red" may be "orange" to someone else. Colors can also look vastly different when compared against two different colors.
Have you tried using the different color pickers in Photoshop? If you click the
a radio button (and set a to 127) in the Photoshop Color picker, you can get a nice selection of reds:
Perhaps your issue isn't having a large enough gamut, but having too large of a gamut. Photoshop comes with quite a few built in color libraries, such a Pantone Solid Coated. If you click
Color Libraries, you can flip through the available colors to look for some inspiration: