Modify my 12 0 12 5 ampere transformer
If you want to modify the transformer in the fashion you describe, the coils will need to be connected in parallel, in the same phase.  This means that, if the output only has three connectors, the center tap connection of one of the two output coils will need to be severed from the center tap pin or wire.  This may or may not be possible, depending on the design of the transformer.
If the transformer is fully metal cased, it will have to be opened.  Once the core and taped windings are exposed, you can try carefully peeling back the tape covering the windings in the area of the center tap wire or lug to see if you can distinguish where the two wires for each half of the coil come together, and then cut one of them.
If the transformer has solder type output lugs or PCB mount legs, you have a better chance that the two wires both go to the connector where you can get to them easily.  If the transformer simply has a wire for the center tap coming directly out of the windings of the transformer, then there is a larger chance that the center tap common connection may be buried inside the windings somewhere, making this project likely not worth the trouble.  If your transformer is torroidal, then you should have easy access to both sides of the center tap connection.
Once you have isolated one of the output coils, you will need to identify which of the two center tap wires belongs to which output coil of the transformer, and this can be done with a simple continuity test using a standard DMM.  Once you have identified which set of now-independent wires belong to which coil, you will need to connect the severed center tap wire to the output of the non-severed coil's non-center tap wire, and connect the output of the severed wire coil's non center-tapped wire to the non-severed coil's center tap output, as per your screenshot.
At this point, the two now-independent coils will be wired in-phase parallel, and you will get half the previous output voltage at double the current capacity.