We improve the state-of-the-art proof techniques for realizing various spectra of $\aT$ in order to realize arbitrarily large spectra. Thus, we make significant progress in addressing a question posed by Brian in his work~\cite{Brian_2021}. As a by-product, we obtain many complete subforcings and an algebraic analysis of the automorphisms of the forcing which adds a witness for the spectrum of $\aT$ of desired size.