Yes, the entire homotopy-type has been computed in a sense.  Although the homotopy-type is described via a chain of fiber-bundles. 

The space $Emb(S^1, \mathbb{R^3})$ has the homotopy-type of a bundle

$$ (C \rtimes K_{3,1}) \times_{SO_2} SO_3$$

where $K_{3,1}$ is the space of embeddings of $I$ into  $D^3$ that agree with a fixed linear embedding on the end points. $K_{3,1}$ is sometimes called the space of long-knots.  The action of $SO_2$ is the rotations in the plane orthogonal to the linear axis. 

$C \rtimes K_{3,1}$ is the tautological knot-exterior bundle over $K_{3,1}$ i.e. it consists of pairs of points $(p,f)$ with $f \in K_{3,1}$ and $p$ in the exterior $D^3 \setminus img(f)$. 

The above result appears in my "Family of embedding spaces" paper, proposition 2.2 (on the arXiv version). The map giving the homotopy-equivalence is quite natural.  You extend elements of $K_{3,1}$ to be embeddings $\mathbb{R} \to \mathbb{R^3}$, extending by the linear function, then apply stereographic projection to convert it to an embedding $S^1 \to S^3$ that agrees with a fixed linear-embedding on a prescribed arc. The point in $C$ you inherits the standard framing from $\mathbb R^3$, and we project that framed point to be in the knot exterior in $S^3$.  Now you do stereographic projection from that framed point to get an embedding $S^1 \to \mathbb{R^3}$.  The $SO_2$ action comes from the fact that one can rotate the knot, or use the rotations of $\mathbb R^3$ to get the same knot. 

The homotopy-type of $K_{3,1}$ has been the subject of a few papers. A brief description appears in the "Family..." paper.  The most compact description I know of appears in "An operad for splicing" and a longer but more basic description appears in "Topology of spaces of knots in dimension 3".  Both can be found on the arXiv.  The components of $K_{3,1}$ have the homotopy-type of compact manifolds.  All the components are finitely-covered by a product of configuration spaces in the plane, but often the cover needs to be quite big.  These manifolds include things like Klein bottles, more generally a class of Bieberbach manifolds, plus configuration spaces in the plane and various twisted products. 

There is a similar description of the homotopy-type of $Emb(S^1, S^3)$ in the "Family..." paper.