Classification of sections of free loop fibration over the two-sphere
The space of sections of $\Lambda S^2\to S^2$ is a subspace of the space of all maps $$S^2\to \Lambda S^2=Map(S^1,S^2).$$ When viewed as a subspace of the space of all maps $$S^1\to Map(S^2,S^2),$$ it becomes $\Omega_1 Map(S^2,S^2)$, the based loopspace of the space of self-maps of $S^2$, with the identity map as basepoint. Thus the problem is to compute $\pi_1(Map(S^2,S^2),1)$. I claim that it has order two.
The space of all degree one maps $S^2\to S^2$ is sometimes denoted by $SG(3)$; it is the "structure group" for oriented spherical fibrations of rank $3$. It is part of a fibration sequence $$ SF(2)\to SG(3)\to S^2 $$ with $SF(2)$ the space of degree one based maps $S^2\to S^2$, a component of $\Omega^2S^2$. Mapping into this is another fibration sequence $$ SO(2)\to SO(3)\to S^2. $$ Thus $SG(3)/SO(3)$ is homotopy equivalent to $SF(2)/SO(2)$.
The latter is $1$-connected and has finite $\pi_2$. This follows as soon as one knows that $\pi_1SO(2)\to \pi_1SF(2)=\pi_3S^2$ is an isomorphism, which I think is not hard to see.
Thus $\pi_1SG(3)$ is the cokernel of the map $\pi_2(SG(3)/SO(3))\to \pi_1SO(3)=\mathbb Z/2$.
This map is zero because it factors through a map $\pi_2(SG(3)/SO(3))\to \pi_1SO(2)$ from a finite group to $\mathbb Z$.
Two sections are fiber wise homotopic if the adjoint maps $S^1\times S^2\to S^2$ are homotopic rel base point$\times S^2$. Cut $S^1$ open at the base point, then you obtain maps $I\times S^2\to S^2$ which are the identity of $S^2$ at each endpoint of the interval $I$. For a homotopy between two such maps $f_1,f_2$ rel endpoints $\times S^2$, we have to extend a map $\partial (I^2)\times S^2\to S^2$ to $I^2\times S^2$.
So we need to extend over a 2-cell $I^2\times pt$, where there is no obstruction, but a choice in $\pi_2S^2=\mathbb Z$, and over the top-dimensional cell where the obstruction is an element in $\pi_3 S^2=\mathbb Z$.
I guess one would now need to understand how this element $o(f_1,f_2,g_{12})$ depends on the choice of extension $g_{12}$ over the 2-cell.
If one has overcome this difficulty, one has an addition formula $$o(f_1,f_2,g_{12})+o(f_2,f_3,g_{23})=o(f_1,f_3,g_{12}\cup g_{13}),$$ see for example Baues, LNM 628, Thm 4.2.7. This could lead to a classification.
Some comments.
1) The space of sections of $LX \to X$ is a grouplike topological monoid, by pointwise multiplication. So, its path components do form a group. In fact it is a loop space $\Omega_1F(X,X)$ of the space of functions $F(X,X)$, where loops are based at the identity map $1\!\!\! : X \to X$.
2) For $A\to X$ a map, let $\Gamma(LX|A)$ denote the space of sections of $LX \to X$ along $A$. This is the same as the space of sections of the pullback $A \times_X LX \to A$. This is also a loop space, this time of $F(A,X)$.
When $A = X$ we set $\Gamma(LX) = \Gamma(LX|A)$.
3) The restriction map $$ F(X,X) \to F(A,X) $$ is a fibration and it loops to the restriction map $\Gamma(LX) \to \Gamma(LX|A)$ . Its fiber at the constant map $A\to X$ is the based function space $F_*(X/A,X)$.
4) Let's apply the above to the inclusion $\ast \subset S^2$ basepoint. This gives a homotopy fiber sequence $$ \Omega^2 S^2 \to F(S^2,S^2) \to S^2\, , $$ where the second map is given by evaluation.
5) Take $\pi_*$ (with respect to the correct basepoint "1") to get an exact sequence of groups $$ \cdots \to \pi_2(S^2) \to \pi_1(\Omega^2S^2;1) \to \pi_1(F(S^2,S^2);1) \to 1 $$ where $\pi_1(F(S^2,S^2);1) = \pi_0(\Gamma(LS^2))$, and $\pi_2(S^2) = \Bbb Z = \pi_1(\Omega^2S^2;1)$. It follows that there's a short exact sequence of groups $$ \Bbb Z \to \Bbb Z \to \pi_0(\Gamma(LS^2)) \to 1\, . $$ It therefore suffices to compute the homomorphism $\Bbb Z \to \Bbb Z$. To compute this, you'll need to understand how the boundary operator works. This amounts in the end to the computation discussed by "user95545" above.