Hybrid functional calculations using different approach
In hybrid functional such as B3LYP, PBE0, HSE etc, some part of HF
  exchange and/or correlation is added to some part of DFT X/C part in
  order to get better results.


First, there is no correlation in HF, only exchange, so you can not add part of the HF correlation into DFT, as you said. In principle, of course, if you want to add not only exchange from the wave function theory (WFT) methods but also some correlation, you could use other that the HF sources, but the conventional hybrid DFT corresponds to mixing in just (a part of) the HF exchange.

Now, you very question of 


  why don't we just use the full HF solution that we already have in our
  hand?


is addressed in almost any textbook on DFT. I suggest that you read, for instance, Section 6.6 Hybrid Functionals in Koch &amp; Holthausen, specifically, pages 78-80. The authors provide a relatively simple argument for for the well-known failure of the exact exchange/density functional
correlation combination approach.

In short, the source of the problem is that the separation of an exchange-correlation entity (would it be energy, or hole, or something else) into exchange and correlation contributions is artificial. Only the whole exchange-correlation entity has a physical meaning, while the above mentioned individual contribution are introduced just for convenience. Usually, it means that exchange and correlation contributions has to be chosen consistently so that their sum, an exchange-correlation entity, has some desired properties.

For instance, the exchange-correlation hole in molecular systems is usually relatively localized, so that the exchange hole and the correlation hole must both be either localized or delocalized (in a very special way so that combined they result in a localized hole). In LDA- and GGA-DFT both the exchange and the correlation hole are localized, but when we mix HF exchange and LDA/GGA-DFT correlation in the most straightforward way, i.e., 
$$
E_{\mathrm{XC}} = E_{\mathrm{X}}^{\mathrm{HF}} + E_{\mathrm{C}}^{\mathrm{DFT}} \, ,
$$
then, I quote:


  We combine the exact, delocalized exchange hole with a localized model
  hole for correlation. Because the cancellation between the two
  individual holes cannot take place, the resulting total hole has the wrong characteristics.




1) Koch, W., &amp; Holthausen, M. C. (2001). A Chemist's Guide to Density Functional Theory, Second Edition, Wiley-VCH Verlag GmbH.