This paper presents an approach to localize a UAV in indoor environments using only a quasi-taut tether. In indoor GPS-denied environments, UAV localization usually depends on vision-based methods combined with inertial sensing, such as visual odometry or SLAM using 2D/3D cameras or laser range finders. This necessitates either heavy and sophisticated sensor payload mounted onto the UAV platform or computationally expensive algorithms running online. In this work, we investigate another indoor localization possibility for a tethered UAV: using the tether's sensory feedback, which is fed into a catenary-based mechanics model, to localize the UAV in an indoor global frame defined by the tether reel center. Our localization method is tested on a physical robot, Fotokite Pro. Our approach could reduce the error of the state-of-the-art tether-based indoor aerial vehicle localization by 31.12%. Since the UAV is localized with respect to the tether reel center, our method could be used to localize the UAV in a moving frame. So it is particularly suitable for inter-localization within marsupial heterogeneous robotic teams for urban search and rescue purposes.