Want to put a new fork on my old road bike.. it currently has an integrated headset with a carbon fork/alloy steer tube/steel crown race.

New fork is carbon/carbon steer tube/integrated crown race. Steer tube is straight, not tapered.

Do I need a new headset, or do I just install the new fork with the old headset and the integrated race just takes the place of and performs the same function as the old steel race? If I do need a new headset, how do I figure out what kind?