Bridge the link by investing in a good DAC/DAP

Digital-to-Analog Converters exist in practically every device that plays audio files, but the ones that come with your phone, tablet or laptop generally aren’t very good. Good DACs don’t have to cost and arm and a leg, and some come with amplifier functions builtin, which will color the sound slightly to suit your preference, and also allow you to use higher-end headphones.

Of course, this means you’ll have to spend some time testing out different DACs (and/or amplifiers) to find which one suits you best, but that’s all part of the fun. Alternatively, you could look at a dedicated Digital Audio Player, as this would come with a quality DAC and an amplifier to start. Of course, we have some recommendations to get you started.