  Consider the classical problem of rational simultaneous approximation to a
point in $\mathbb{R}^{n}$. The optimal lower bound on the gap between the
induced ordinary and uniform approximation exponents has been established by
Marnat and Moshchevitin in 2018. Recently Nguyen, Poels and Roy provided
information on the best approximating rational vectors to points where the gap
is close to this minimal value. Combining the latter result with parametric
geometry of numbers, we effectively bound the dual linear form exponents in the
described situation. As an application, we slightly improve the upper bound for
the classical exponent of uniform Diophantine approximation
$\widehat{\lambda}_{n}(\xi)$, for even $n\geq 4$. Unfortunately our
improvements are small, for $n=4$ only in the fifth decimal digit. However, the
underlying method in principle can be improved with more effort to provide
better bounds. We indeed establish reasonably stronger results for numbers
which almost satisfy equality in the estimate by Marnat and Moshchevitin. We
conclude with consequences on the classical problem of approximation to real
numbers by algebraic numbers/integers of uniformly bounded degree.
