Transitive permutation groups which all of their proper subgroups are intransitive

Consider the symmetric group $S$ on $n$ symbols in its action on the $k$-subsets of $\{1,\ldots,n\}$. If $k\ne2,4$ and $n=2k+1$, then the only transitive subgroups of $S$ are $S$ itself and the alternating group. Hence in this case the alternating group acts transitively but all its proper subgroups are intransitive. See my "More Odd Graph Theory", Discrete Math. 32 (1980) 205-207. The basic idea is that a group that is transitive on $k$-subsets must be $(k-1)$-transitive and in most cases this leaves with only the symmetric and alternating groups.

As Geoff Robinson has already written in his comment, you are asking for minimally transitive permutation groups. Specifically you ask whether there is any class of such groups other than the regular permutation groups. The answer to this is a clear yes. -- For example there are up to conjugacy only $51$ regular permutation groups of degree $32$, but there is a total of $11605$ conjugacy classes of minimally transitive permutation groups of degree $32$. See Page 3 of the paper The Transitive Permutation Groups of Degree 32 by Cannon and Holt which reports about the determination of all $2801324$ conjugacy classes of transitive permutation groups of degree $32$.

Whenever one has a vertex-transitive graph which is not a Cayley graph, one has an example of such a group (maybe more than one). E.g. both $A_5$ and $5:4$ act vertex-transitively on the Petersen graph, and are minimal transitive subgroups of $S_{10}$. See e.g. the paper by B.D.McKay and C.E.Praeger.