  Helical trilayer graphene (hTG) exhibits a supermoir\'e pattern with large
domains centered around stacking points ABA and BAB, where two well-separated
low-energy bands appear with different total Chern numbers at each valley,
forming a Chern mosaic pattern. In the chiral limit, the low-energy bands
become exactly flat at zero energy for magic-angle twists. Here we investigate
these zero-energy flat bands and their topological properties in the presence
of a perpendicular magnetic field. We show that hTG retains the precise
flatness of the zero-energy bands, even at finite magnetic fields. We find
topological phase transitions at fields corresponding to unit and half magnetic
flux leading to an emergence of higher Chern number flat bands. Consequently
the Chern mosaic gets modified for finite magnetic fields. We further find the
analytical forms of zero-energy wave functions and identify a set of hidden
wave functions, which gives crucial insights into both the topological
transitions and enhancement of Chern numbers across them. We also find
topological transitions away from the chiral limit with finite corrugations and
at different magic angles.
