I don't know of any general construction. Perhaps the most 'natural' examples are loop transversals: Given a left transversal $X$ to $H$ in $G$ with $1 \in X$, define a binary operation on $X$ by setting $x*y$ to be the unique element of $X \cap xyH$. Then $*$ is right-cancellative; if $X$ is in fact a left transversal to every conjugate of $H$, then $*$ is also left-cancellative and hence $X$ forms a loop.

I don't know who thought of this idea first, but I found a series of articles by E.A. Kuznetsov on the subject, for example: http://www.quasigroups.eu/contents/download/1999/6_1.pdf

I don't know of any general construction. Perhaps the most 'natural' examples are loop transversals: Given a left transversal $X$ to $H$ in $G$ with $1 \in X$, define a binary operation on $X$ by setting $x*y$ to be the unique element of $X \cap xyH$. Then $*$ is left-cancellative; if $X$ is in fact a left transversal to every conjugate of $H$, then $*$ is also right-cancellative and hence $X$ forms a loop.

I don't know who thought of this idea first, but I found a series of articles by E.A. Kuznetsov on the subject, for example: http://www.quasigroups.eu/contents/download/1999/6_1.pdf