Given a pointed space X, let $J(X)$ denote its James construction. There is a natural inclusion $X\rightarrow J(X)$ which can equivalently be described as the unit map $\eta_X:X\rightarrow \Omega \Sigma X$. The homotopy fiber of the counit map $\epsilon_X:\Sigma \Omega X\rightarrow X$ is known to be $\Omega X * \Omega X$. Is there an analogous result for the homotopy fiber of $\eta_X$ in terms of $X$?

This might not be a satisfactory answer, but let me point out that S.c. Wong constructed a "little cube" model for the homotopy fiber of $\Omega^{k1}E^n : \Omega^{k1}\Sigma^{k}X \to \Omega^{k1+n}\Sigma^{k+n} X$ in this paper in 1994. Note that May's little cube model for $\Omega\Sigma X$ is essentially the James construction. 

