In dimension 2, Edmonds proved that every degree one map between closed orientable surfaces is homotopic to a "pinch map", which takes a subsurface which is bounded by one component, and sends it to a disk. This is a special case of the connect sum method you describe, so that is really the only method in 2D.
In dimension 3, Yi Liu has proven that every closed orientable 3-manifold has a degree one map to finitely many other closed orientable 3-manifolds. I think it ought to be possible to make the proof into an algorithm. So given a closed manifold $M$, there should be an algorithm which produces finitely many manifolds $N$ such that there is a degree one map $M\to N$. It might also be possible to say something about the homotopy classes of such maps, but this hasn't been worked out. I should say also that there are many known 3-manifolds which have non-zero degree maps only to themselves or $S^3$.
Unfortunately, the proof doesn't give a prescription of how to produce such maps. There is a version of the pinching map for Seifert manifolds, by extending a pinch over the base orbifolds to the fibrations.
A related method for producing degree one maps is to observe that any knot complement has a degree one map to a solid torus which is the identity on the boundary torus. Thus, one can glue the knot complement and the solid torus to another manifold with torus boundary, and extend the degree one map as the identity over the manifold. This only works if you have a manifold with essential tori in it.
A third method allows you to specify the range, but not the domain. Take a 3-manifold $N$, and take a knot $K\subset N$ which is null-homotopic. Then one may perform a surgery on this knot with slope $\alpha$ to get a manifold $N_K(\alpha)$ which has a degree one map $N_K(\alpha)\to N$, by extending the meridian of the Dehn filling over a disk in $N$ which realizes the null-homotopy of $K$. This method is described in Section 3 of this paper.
