Why do you re-wind the secondary?
Transformers are fairly simple devices, you have a metal core, typically shaped like EI, they are called E-I cores because they are made of a bunch of E and I shaped thin slabs of magnetic-firendly material stacked together. Then you have the windings, which are enameled wire (insulated with a thin coating) wrapped around the center leg of the E in the EI shape. There are usually two windings, a primary winding and a secondary winding, but you can have as many secondaries as you'd like. The primary winding is hooked up directly to the wall, so you don't want to touch that winding - it is already made to safety standards and set up. The reason you want to unwind the secondary and put a new one on is you will NOT find what you are looking for in any transformer on the market. There is no transformer commercially available to do exactly what you want (there is one
made for pedalboard supplies but it is only nine outputs for 9 volts I believe, it's called the WPDLXFMR-1), since in your case you would need twelve secondary windings at ~10VAC... So the solution is to get a transformer, take off whatever secondary it has on it, and wind twelve different secondary windings onto the core to get your twelve 10VAC outputs.
The transformer outputs a voltage based on a ratio of turns between windings. For example, a 120:1 turns ratio transformer will output 1VAC on the secondary when 120VAC is put into the primary. To get the 10VAC outputs with an already made primary, you wind say 10 turns onto the secondary, reassemble the core, and test the voltage on the output. Using the voltage you get on the test and knowing the number of turns it took to get that voltage, you calculate how many turns you need to get to 10VAC or whatever voltage you need (aim for a peak voltage 2-3 volts higher then the output you want to have, and then use a regulator to smooth it out and bring it down). THen begin the laborious process of winding them all.