For the circle there's an answer. In fact there's a whole chapter in the book
Burghelea, Dan; Lashof, Richard; Rothenberg, Melvin: Groups of automorphisms of manifolds. With an appendix ("The topological category'') by E. Pedersen. Lecture Notes in Mathematics, Vol. 473. Springer-Verlag, Berlin-New York, 1975.
dedicated to showing that after looping once, the space $\text{Diff}(M\times S^1)$ splits up to homotopy as (the loops of) $$ \text{Diff}(M) \times B\text{Diff}(M) \times \eta(M) , $$ where the middle term is a non-connective one-fold delooping of $\text{Diff}(M)$ and $\eta(M)$ is the mysterious "nil-term." In particular, once gets a decomposition on the level of homotopy groups. (This theorem is an analog of the Bass-Heller-Swan type result which says $K(R[t]) \simeq K(R) \times BK(R) \times \eta(R)$.)
One can say something about the homotopy type the nil-term in the the concordance stable range, roughly, $\dim M/3$.