Which AT cartridge do you have? MM cartridge gain is measured in milli volts (mV), usually running from 2.5 to 5 mV. Phonostage gain is measured in decibels (dB), Most budget MM ones usually run from 34 to 50 dB. Do you know the gain of yours? Knowing these values would help to derive the best cartridge/phonostage combination to achieve your goal.
A doubling of voltage provides a 6 dB increase in gain. A 9-10 dB gain is required to provide a doubling of perceived loudness. The output of your CD player is likely 2.something volts. Your cartridge/phonostage is likely putting out ~1 volt. Therefore the CD player provides ~6dB of gain over your cartridge/phonostage combo.