2D drawing, making 2d polygone 1 texel more thick to get rid of border noise

I use alpha blending to draw onto a very low resolution render target.
A median filter helps to get rid of noise from the texture. But the polygone borders are still a problem.

What I basicly would like to have is a way to “thicken” my 2d polygone. So that in all 4 directions one more texel is occupied and drawen to.

I guess there is no offset I could change how the texels are picked by the rasterizer?

I didn’t use it so far, but the geometry shader looks like it could be used for this by changing the vertex positions and then correcting the texture coord?