  We propose the notions of uniform local weak o-minimality and $*$-local weak
o-minimality. Local monotonicity theorems hold in definably complete locally
o-minimal structures and uniformly locally o-minimal structures of the second
kind. In this paper, we demonstrate new local monotonicity theorems for
uniformly locally weakly o-minimal structures of the second kind and for
locally o-minimal structures under the assumption called the univariate
$*$-continuity property. We also prove that several formulas for dimension of
definable sets which hold in definably complete locally o-minimal structures
also hold in $*$-locally weakly o-minimal structures possessing the univariate
$*$-continuity property.
