  In this work we aim to investigate non-mainstream thick tori configurations
around Kerr Black Holes with Scalar Hair (KBHsSH). For that goal, we provide a
first approach using constant specific angular momentum non-self-gravitating
Polish doughnuts. Through a series of examples, we show the feasibility of new
topologies, such as double-centered tori with two cusps as well as similar
structures as the ones found for rotating Boson Stars (BSs), namely tori
endowed with two centers and a single cusp. These KBHsSH' solutions are also
shown to possibly house static surfaces, associated to the static rings present
in these spacetimes. Through this report we highlight the differences between
these fluid configurations when housed by some KBHsSH examples, standard Kerr
black holes and rotating BSs.
