The activated escape of an underdamped Brownian particle out of a deep potential well is characterized by weak friction \ensuremath{\gamma}\ensuremath{\ll}\ensuremath{\omega} (\ensuremath{\gamma} is the coefficient of friction and \ensuremath{\omega} is a typical frequency of the intrawell motion) and by a large barrier height ${\mathit{U}}_{0}$\ensuremath{\gg}T (${\mathit{U}}_{0}$ is the barrier height and T is the temperature). The approach developed previously to calculate the decay rate is based on the derivation of an integral equation and enables one to sum up an infinite series in powers of the ratio \ensuremath{\gamma}${\mathit{U}}_{0}$/T\ensuremath{\omega}\ensuremath{\sim}1 contributing to the preexponential factor of the Arrhenius law. In the present paper it is shown that the leading correction to the above result comes from the slowing down of the particle motion near the top of the barrier and is of the order of (T/${\mathit{U}}_{0}$) ln(${\mathit{U}}_{0}$/T). To calculate it explicitly, one needs to find a correction to the kernel of the above-mentioned integral equation. Beyond the leading-logarithmic approximation, two different factors contribute corrections of the order of T/${\mathit{U}}_{0}$\ensuremath{\sim}\ensuremath{\gamma}/\ensuremath{\omega}. The noise-induced effects in the barrier crossing-recrossing by particles in a narrow energy range \ensuremath{\varepsilon}\ensuremath{\sim}\ensuremath{\gamma}T/\ensuremath{\omega} can be easily incorporated into the general scheme of the calculations. On the other hand, a more accurate derivation of the kernel of the integral equation is required to take into account small variations of the intrawell particle motion caused by variations of the particle energy on the scale T\ensuremath{\ll}${\mathit{U}}_{0}$ under the effects of friction and thermal noise. The proposed consistent expansion in terms of the small parameters of the problem provides an effective approach to a quantitative investigation of the turnover behavior in the Kramers problem. For the regime of an intermediate-to-strong friction, the finite-barrier corrections can be neglected, since, for typical barrier shapes, they are always small.