The origin of the exchange bias between a ferro- and an antiferromagnet still is not fully resolved. Many structural parameters such as film thickness, roughness, domain size, domain walls, strain, dislocations, and surface defects play a role. To disentangle their individual contributions, one must correlate each of these structural effects with the exchange bias. Here we report our first such attempts using thin films of the antiferromagnetic orthoferrite LaFeO3 grown by molecular beam epitaxy. We investigate how the epitaxial relationship and strain affect the domain configuration as well as their size and orientation. Next, we explore how these different domains can pin ferromagnetic domains in exchange-coupled systems. We conclude that decreasing the domain size of the antiferromagnet leads to a higher exchange bias. The highest exchange bias was obtained for LaFeO3 films on MgO substrates.