  Let $Y$ be the space obtained by attaching a finite-type wedge of cells to a
simply-connected, finite-type CW-complex.
  We introduce the free and semi-inert conditions on the attaching map which
broadly generalize the previously studied inert condition. Under these
conditions we determine $H_*(\Omega Y;R)$ as an $R$-module and as an
$R$-algebra respectively. Under a further condition we show that $H_*(\Omega
Y;R)$ is generated by Hurewicz images.
  As an example we study an infinite family of spaces constructed using only
semi-inert cell attachments.
