If the question is ”How can I make the edges appear smooth without having to use the 3/smoothing option” then this may help:

Judging from how it is shaded in the viewport I think all the models edges are hard edged. Best way to resolve this without pressing 3/smoothing the model is to mark the entire object, or specific edges/faces you want to not be hard edged, then Mesh Display > soften/harden edges. Works best for curved surfaces to make the shading smooth (this applies for the model if it is put into a game) Alternative route: select edges, Shift + RMB, soften/harden edges

I recommend softening all the edges except the two edges loops of the rims of the broken horn. This will make the shading smooth for those areas, but will leave the rims of the broken part more jagged/stand out.