Magnetic topological phases of quantum matter are an emerging frontier in physics and materials science, of which kagome magnets appear as a highly promising platform. Here, we explore magnetic correlations in the recently identified topological kagome system TbMn₆Sn₆ using muon spin rotation, combined with local field analysis and neutron diffraction. Our studies identify an out-of-plane ferrimagnetic structure with slow magnetic fluctuations which exhibit a critical slowing down below T∗C1 ≃ 120 K and finally freeze into static patches with ideal out-of-plane order below T_C1 ≃ 20 K. We further show that hydrostatic pressure of 2.1 GPa stabilises the static out-of-plane topological ferrimagnetic ground state in the whole volume of the sample. Therefore the exciting perspective arises of a magnetically-induced topological system whose magnetism can be controlled through external parameters. The present results will stimulate theoretical investigations to obtain a microscopic understanding of the relation between the low-temperature volume-wise magnetic evolution of the static c-axis ferrimagnetic patches and the topological electronic properties in TbMn₆Sn₆.