The aim of this work is to study a very special family of odd-quadratic Lie superalgebras \( \mathfrak{g}=\mathfrak{g}_{\overline 0}\oplus \mathfrak{g}_{\overline 1} \) such that \( \mathfrak{g}_{\overline 1} \) is a \( \mathfrak{g}_{\overline 0} \)-module (weak filiform type). We introduce this concept after having proved that the unique non-zero odd-quadratic Lie superalgebra (\( \mathfrak g \),B) with \( \mathfrak{g}_{\overline 1} \) a filiform \( \mathfrak{g}_{\overline 0} \)-module is the abelian 2-dimensional Lie superalgebra \( \mathfrak{g}=\mathfrak{g}_{\overline 0}\oplus \mathfrak{g}_{\overline 1} \) such that \( \mathrm{rm}\ \mathrm{dim}\mathfrak{g}_{\overline 0}= \mathrm{rm}\ \mathrm{dim} \mathfrak{g}_{\overline 1} =1 \). Let us note that in this context the role of the center of \( \mathfrak g \) is crucial. Thus, we obtain an inductive description of odd-quadratic Lie superalgebras of weak filiform type via generalized odd double extensions. Moreover, we obtain the classification, up to isomorphism, for the smallest possible dimensions, that is, six and eight.