  In this paper, we study a relationship between tilting modules with finite
projective dimension and dominant dimension with respect to injective modules
as a generalization of results of Crawley-Boevey-Sauter,
Nguyen-Reiten-Todorov-Zhu and Pressland-Sauter. Moreover, we give
characterizations of almost $n$-Auslander-Gorenstein algebras and almost
$n$-Auslander algebras by the existence of tilting modules. As an application,
we describe a sufficient condition for almost $1$-Auslander algebras to be
strongly quasi-hereditary by comparing such tilting modules and characteristic
tilting modules.
