Here is a "Scheme" of the setup I had by the time I started this thread:
Turntable ----> RM-4 ----> Headphone Amp
As I described earlier the problem was in a very low sound level at the headphones even while the volume knob dialed to its absolute maximum position.
The solution was is to place another amplifier that had PHONO IN/OUT integrated.
So, here is what a scheme of the system that works now:
Turntable ----> RM-4 ----> Integrated Amplifier With Phono In/Out built ----> Headphone Amp
I have to say guys, that this particular setup does sound great!!!.... I am all happy fellow now

Here are the elements used:
Pro-ject RPM 5 SE Turntable with Blue Point V.2 cartridge
Music Reference RM-4 phono preamp
DK Design Group MK integrate amplifier (with PHONE In/Out integrated) (it also has tube preamp and solid stage amp).
HeadRoom Max headphone amplifier driving Denon HD5000
The thing I'm concerned about is the size of the DK Design amplifier used here purely because of its Phono In/Out jacks. I wonder if there could be a more elegant alternative/solution than using a full size (HUGE and HEAVY) integrated amp... may be some "stand alone" phone preamplifier would do it?.... strange I always thought that Music Reference RM-4 IS indeed a phono pre-amplifier... If Music Reference RM-4 is not Phono-Preamp, then what is it? And why I can use it only with the amp that has a phono stage in it? What do I have to do IF the amp doesn't have PHONO-IN (like my headphone amplifier)?