This article concerns an extension of the topological derivative concept for 3D inverse acoustic scattering problems, whereby the featured cost function $J$ is expanded in powers of the characteristic size $\epsilon$ of a sound-hard scatterer about $\epsilon=0$. The $O(\epsilon^{6})$ approximation of $J$ is established for a small scatterer of arbitrary shape of given location embedded in an arbitrary acoustic domain, and generalized to several such scatterers. Simpler and more explicit versions of this result are obtained for a centrally-symmetric scatterer and a spherical scatterer. An approximate and computationally fast global search procedure is proposed, where the location and size of the unknown scatterer is estimated by minimizing the $O(\epsilon^{6})$ approximation of $J$ over a search grid. Its usefulness is demonstrated on numerical experiments, where the identification of a spherical, ellipsoidal or banana-shaped scatterer embedded in a acoustic half-space from known acoustic pressure on the surface is considered.