In an answer to this question, I learned the following from Tyrone, citing G. Lang's The evaluation map and the EHP spectral sequence.
Let $(X,x)$ be a pointed connected space, and let $\lambda \in LX$ be a lift of the basepoint -- equivalently, $\lambda \in \Omega X$. Then we have a pointed fibration sequence
$$(\Omega X, \lambda) \to (LX, \lambda) \to (X,x)$$
and an induced long exact sequence in homotopy. To a large extent, then, your question boils down to determining what the boundary map $\pi_{n+1}(X) \to \pi_{n}(\Omega X)$ is in the long exact sequence. As you've anticipated, this map is not the canonical isomorphism between these groups. According to Lang, this boundary map is the Whitehead product with $\lambda$, i.e. it's $[-,\lambda]: \pi_{n+1}(X) \to \pi_{n+1}(X)$.
When $n=0$, this comes to saying that the map is $\alpha \mapsto \alpha \lambda \alpha^{-1}$, and we recover the fact that $\pi_0(LX)$ is in canonical bijection with the set of conjugacy classes in $\pi_1(X)$. The kernel of the map with basepoint $\lambda$ is the centralizer $Z_{\pi_1(X)}(\lambda)$.
So to partially answer your question, we have that $\pi_1(LX,\lambda)$ sits in a short exact sequence
$$0 \to \operatorname{coker}(\pi_2(X) \xrightarrow{[-,\lambda]} \pi_2(X)) \to \pi_1(LX,\lambda) \to Z_{\pi_1(X)}(\lambda) \to 0$$
and more generally, we have a short exact sequence
$$0 \to \operatorname{coker}(\pi_{n+1}(X) \xrightarrow{[-,\lambda]} \pi_{n+1}(X)) \to \pi_n(LX,\lambda) \to \operatorname{ker}(\pi_{n}(X) \xrightarrow{[-,\lambda]} \pi_{n}(X)) \to 0$$
But I don't know anything about describing the relevant extension. I think that the relevant action of $Z_{\pi_1(X)}(\lambda)$ on $\pi_2(X)/[-,\lambda]$ is at least the usual action of $\pi_1$ on $\pi_2$, but I'm not really sure.
It is hard to say what Hatcher really means, unfortunately he isn't always precise in his formulations.
As far as I know we need a pointed fibration to get the eaxct sequence in your question. However, if you want to work with free fibrations $p : E \to B$, you can easily prove that for each well-pointed $(X,x_0)$ you get an exact sequence
$$\langle (X,x_0), (F,e_0)\rangle \to \langle (X,x_0), (E,e_0)\rangle \to \langle (X,x_0), (B,b_0)\rangle$$
I guess this is what Hatcher has in mind since he mainly works with CW-complexes in Chapter 4 (he does not say anything explicitly about $X$).
Best Answer
Usually, the result will be $\Sigma \Omega X$ (the answer for $S^1$ should actually be an infinite wedge of circles), this is not particularly special to path and loop spaces, but rather to pairs of spaces where the total space is contractible.
If $X$ is a based CW complex or manifold $\Omega X \hookrightarrow PX$ should be a cofibration; I am not sure where to find a reference for this fact, but maybe in some of Milnor's work on the homotopy type of function spaces between CW complexes.
Anyways, this means that $PX/\Omega X \simeq PX \cup \operatorname{cone}(\Omega X)$. The latter maps to $\Sigma (\Omega X)$ which we model as $\operatorname{cone}(\Omega X)/(\Omega X) \times \{0\}$ by collapsing $PX$ to a point. Again, in reasonable circumstances like $X$ a CW complex or manifold, this map will be a homotopy equivalence by work of Milnor. I am not sure if it is true in general, it is really a question about if one needs to take CW approximations before gluing in cones to calculate homotopy pushouts.