An intriguing question in the context of dynamics arises: Could a moon possess a moon itself? Such a configuration does not exist in the Solar System, although this may be possible in theory. Kollmeier et al. (2019) determined the critical size of a satellite necessary to host a long-lived sub-satellite, or submoon. However, the orbital constraints for these submoons to exist are still undetermined. Domingos et al. (2006) indicated that moons are stable out to a fraction of the host planet’s Hill radius RH,p; which in turn depend on the eccentricity of its host's orbit. Motivated by this, we simulate systems of exomoons and submoons for 105 planetary orbits, while considering many initial orbital phases to obtain the critical semimajor axis in terms of RH,p or the host satellite's Hill radius RH,sat, respectively. We find that, assuming circular coplanar orbits, the stability limit for an exomoon is 0.40RH,p and for a submoon is 0.33RH,sat. Additionally, we discuss the observational feasibility of detecting these sub-satellites through photometric, radial velocity, or direct imaging observations using the Neptune-sized exomoon candidate Kepler 1625b-I and identify how stability can shape the identification of future candidates.