A fully self-consistent hybrid dRPA (direct random phase approximation) method, named sc-H[γ]dRPA, is presented with γ = 1/3. The exchange potential of the new method contains a fraction γ of nonlocal Hartree-Fock-like exchange besides the exact local Kohn-Sham (KS) exchange potential. The sc-H[γ]dRPA method, in contrast to a straightforward self-consistent dRPA method within the KS formalism, does not suffer from convergence problems for systems with small eigenvalue gaps. Moreover, the sc-H[γ]dRPA method yields distinctively more accurate reaction, isomerization, and transition state energies than other dRPA approaches, e.g., the frequently used non-self-consistent dRPA method using orbitals and eigenvalues from a KS calculation with the exchange-correlation potential of Perdew, Burke, and Ernzerhof (PBE). The sc-H[γ]dRPA method outperforms second-order Møller-Plesset perturbation theory and coupled cluster singles doubles methods while exhibiting a more favorable scaling of computational costs with system size. A value of γ = 1/3 is shown to be a good choice also for a dRPA@PBE[γ] method, which is a non-self-consistent dRPA method using orbitals and eigenvalues from the hybrid PBE0 method with an admixture of γ = 1/3 of exact exchange instead of the 25% of the PBE0 functional. The dRPA@PBE[γ] method yields reaction, isomerization, and transition state energies that are as good as the sc-H[γ]dRPA ones but is computationally simpler and more efficient because it does not require the self-consistent construction of the dRPA correlation potential. The direct sc-H[γ]dRPA, on the other hand, in contrast to all standard density-functional methods, yields qualitatively correct correlation potentials.