  We determine the submaximal dimensions of the spaces of almost Einstein
scales and normal conformal Killing fields for connected conformal manifolds.
The results depend on the signature and dimension $n$ of the conformally
nonflat conformal manifold. In the Riemannian case, these two dimensions are at
most $n-3$ and $\frac{(n-4)(n-3)}{2}$, respectively. In the Lorentzian case,
these two dimensions are at most $n-2$ and $\frac{(n-3)(n-2)}{2}$,
respectively. In the remaining signatures, these two dimensions are at most
$n-1$ and $\frac{(n-2)(n-1)}{2}$, respectively. This upper bound is sharp and
to realize examples of submaximal dimensions, we first provide them directly in
dimension 4. In higher dimensions, we construct the submaximal examples as the
(warped) product of the (pseudo)-Euclidean base of dimension $n-4$ with one of
the 4-dimensional submaximal examples.
