I suggest, you really look carefully into the reference "Humphreys, Reflection and Coxeter groups" (link behind paywall), as you again find the answer there.
Please look into the reference for details below. In case, you have no access to it, you might be lucky searching for it on the web (you can also drop me a line).

First, one has to define a root system for $W$ (done in Section 5.4) as $\Phi = \{w(\alpha_s) : w \in W, s \in S\}$. Then there is a decomposition $\Phi = \Phi^+ \sqcup -\Phi^+$ with $\Phi^+ = \Phi \cap \mathbb{R}_{\geq 0}\Delta$ for $\Delta = \{\alpha_s : s \in S\}$.
And finally, by construction, $R = \{ s_\beta : \beta \in \Phi^+\}$ and this is a bijective correspondence. So the answer to your question is **yes**.