On a complete Riemannian manifold $(M,g)$ with positive injectivity radius $R$, a geodesic curve $g_1:[a,b]\rightarrow M$ enters a point, $g_1(b_1)=p$ with tangent $v$ and another geodesic curve $g_2:[b,c]\rightarrow M$ exists the point, $g_2(b)=p$ with the same tangent $v$.
Is the following true? There is a parametrisation and some nontrivial interval $[d,e]\subset[a,c]$ and a geodesic curve $g$ such that for some $f\in(d,e)$, $g{[d,f]}\subset g_1[a,b]$ and $g[f,e]\subset g_2[b,c]$.
Essentially, if a geodesic curve $g_1$ interpolates a point $p$ at its endpoint and another $g_2$ interpolates $p$ at its start point, and they both have the tangent $v$ at $p$, then (some nontrivial subset of them) can be combined into a single geodesic curve.
Attempt: I think this follows from local uniqueness of the exponential map at $p$ due to $R>0$.