 Near-surface recoil height with constant downward thrust

With upward initial speed $v$ and a constant downward force $F$ on remaining mass $M$, the near-surface excursion is $H=v^2/[2(g+F/M)]$. A spherical body's surface gravity can be inferred from its escape speed: $g=v_{\mathrm{esc}}^2/(2R)$. The approximation needs $H\ll R$ and no additional tether constraint.