The following definitions are from Comparison Theorems in Riemannian Geometry by Cheeger & Ebin.
Let $(M,g)$ be a Riemannian manifold. The geodesic submanifold defined by a tangent vector $v\in T_pM$ is $\exp_p(B)$ where $B\ni0$ is a small ball in $\operatorname{span}(v)^\perp\subset T_pM$, so small that $\exp_p|_B$ is an embedding. Let $\gamma(t):=\exp_p(tv)$. We consider points focal to the geodesic submanifold defined by $v$ along the geodesic $\gamma$.
In the book there is a Rauch II, i.e., second Rauch comparison, which requires the geodesic to have no focal points in the above sense.
Question. Let $c:[0,\ell]\to M$ be a geodesic and $E$ a parallel field along $c$ orthogonal to $c'$. Does there exist an $\varepsilon>0$ such that for all $t\in[0,\ell]$, the there are no points on the geodesic $$\gamma_t:[0,\varepsilon]\to M,s\mapsto\exp_{c(t)}(sE(t))$$ that is focal to the geodesic submanifold defined by $E(t)$ along $\gamma_t$?
This seems to be implicitly assumed in the proof of a lemma for the Cheeger–Gromoll soul theorem. However, I don't see why this is true. The existence of such a focal point is equivalent to having a critical point of some restriction of $\exp$, but for each $t$ the restrictions are different and don't seem to be related. So I don't know how to make the $\varepsilon$ uniform with respect to $t$.
Any help is deeply appreciated!
Edit: It seems that the notion of focal points is not so well-known. Here it is in the sense as in, e.g., Morse Theory by Milnor. Let $N$ be a submanifold, $\gamma$ a unit-speed geodesic with $\gamma(0)\in N$, $\gamma'(0)\perp T_{\gamma(0)}N$. Let $T^\perp N\subset TM$ be the normal bundle of $N$. A point $\gamma(a)$ is said to be focal to $N$ along $\gamma$ if $\exp_p|_{T^\perp N}:T^\perp N\to M$ is singular at $a\gamma'(0)$ (note that $\exp(a\gamma'(0))=\gamma(a)$). In particular, when $N$ is a single point this reduces to the usual notion of conjugate points along a geodesic.