Standard transformer for higher-voltage dual output flyback converter
An LM2586-12 is a 12V fly-back regulator chip - note pin 3 - it "sniffs" the dc output of the diode/capacitor and when it equals 12 volts the chip starts regulating the pulse width to maintain the output at 12 volts. If the 12V output rises a little high, the chip momentarily stops pushing energy thru the transformer.

Fly back designs are not like conventional transformers - in a fly-back design the circuit pumps energy into the transformer primary in one half cycle and that energy gets released by the secondary on the 2nd half cycle. The energy released produces a voltage that is load dependent and only in extreme cases of duty cycle requirements is the turns ratio altered to get a bigger/smaller output voltage. Typically, a 1:1 turns ratio produces input-output voltages that are the same value for a 50:50 duty cycle but this is load dependent too.

Try looking for an adjustable version of the LM2586 or maybe, as an experiment try potting down the feedback with a potential divider to "con" the chip that it is seeing 12 volts BUT before doing so, read the data sheet to see if they allow this.