This paper introduces two novel strategies for iterative reconstruction of full interior tomography (FINT) data, i.e. when the field of view is entirely inside the object support and knowledge of the object support itself or the attenuation coefficients inside specific regions of interest are not available. The first approach is based on data edge-padding. The second technique creates an intermediate virtual sinogram, which is, then, reconstructed by a standard iterative algorithm. Both strategies are validated in the framework of the alternate direction method of multipliers plug-and-play with gridding projectors that provide a speed-up of three orders of magnitude with respect to standard operators implemented in real space. The proposed methods are benchmarked on synchrotron-based X-ray tomographic microscopy datasets of mouse lung alveoli. Compared with analytical techniques, the proposed methods substantially improve the reconstruction quality for FINT underconstrained datasets, facilitating subsequent post-processing steps.