Can the exponential map be used to define geodesics (and hence, generalisations of geodesics)? Let $(M,g)$ be a (connected, paracompact, $C^{\infty}$-smooth) Riemannian manifold with Riemannian metric $g$. The exponential map is defined for each point $p \in M$ to be the map $\exp_p : T_p M \to M$ that sends a tangent vector $v \in T_p M$ to the endpoint of the unique geodesic $\gamma$ satisfying $\gamma(0)=p$ and $\gamma'(0)=v$.
In particular, the notion of a geodesic allows us to define an exponential map. My question concerns the reversal of this construction:

Suppose that for each $p \in M$, we have a map $\sigma_p : T_p M \to M$ with some desirable properties. Can one define a notion of $\sigma$-geodesic that coincides with notion of a geodesic when $\sigma =\exp$?

If this can be done, I suspect that it is well-known, but I've never come across such a construction.
 A: Exponential map in your definition is closely related to  the smooth family of smooth  curves smoothly depending on the position such that  in every point in every direction there exists precisely one curve at this point in this direction. In literature, such a family is sometimes called    path structure.
There are two  differences between your structure and path structure:  the first one is that you curves are parameterised, and the curves of path structure are not. But this an additional structure, so my answer below will has sense also for your setup.
The other  difference  is that for a path structure there exists   precisely one, or, in the case of irreversible path structures, at most two  curves from  the family. You definition allows that (for three points A,B,C) the curves from A and B to C come  to C with the same velocity vector which is not possible for geodesics. Or, even more wild, take 3 points A, B, C and consider the trajectory $\gamma_{ABC} $
of your exponential map which starts  from A  and then passes through first B and then C (assume such  trajectory exists,  i.e., choose B and C  on a trajectory   from A). Your  definition does not require  that the segment of $\gamma_{ABC}$ from $B$ to $C$ is a trajectory of your exponential map starting from $B$ and going to $C$, though this property may be essential for geodesics.
But, returning to path structures, most path structures do not come from an affine connection (known since at least Cartan  for dim 2, essentially the same proof works  for all dimensions, see https://arxiv.org/abs/1101.2069). In dimension 2, every reversible path structure is locally Finsler-metrisable https://arxiv.org/abs/1002.0243, the question whether every  irreversible path structures is  metrisable is still open.
In higher dimensions most path structures are, even microlocally, not Finsler-metrisable, see (the open access paper) https://link.springer.com/article/10.1007/s10714-022-03006-2 .
A: Let’s define $\gamma:\mathbb{R}\to M$ as a $\sigma$-geodesic iff for any $a,b,c\in\mathbb{R}$, there are vectors $v,w$ in the tangent space at $\gamma(a)$ with
$$\sigma_{\gamma(a)}(v)=\gamma(b)$$
$$\sigma_{\gamma(a)}(w)=\gamma(c)$$
$$(c-a)v=(b-a)w$$
If $\sigma=\exp$, and $\gamma$ is a geodesic, this property clearly holds.
If $\sigma=\exp$, and this property holds for $\gamma$, then let $P(a,k)$ be the statement that $d(\gamma(a),\gamma(b))=k|a-b|$ whenever $\gamma([a,b])$ is entirely within the injectivity radius of $\gamma(a)$. Then there is some $k$ with $P(0,k)$, and for that $k$, the set $\{a:P(a,k)\}$ is both open and closed in $\mathbb{R}$. So in fact $P(a,k)$ holds for all $a$, $d(\gamma(a),\gamma(b))=k|a-b|$ for all sufficiently close $a$ and $b$, and therefore $\gamma$ is a geodesic.
