How to create mobile version of existing site?

Hi there, looking for a bit of advice more than anything.

I have a desktop version of a site here, that they now want to be a mobile version of.

charlesregionalcareers.org

I’ve built sites in responsive manners before, but the thing is that the layout they are desiring in the mobile version looks so much different than the desktop that it would make sense to just create a mobile only template of the page… is it possible to have it so that if someone visited that site in a mobile device, it would then bring up a mobile only page that’s redirected from the desktop version?

Any guidance would be appreciated