Here is the all the Unbound Remix New Custom Car unlock condition:

Unlock by default:
> Ford Mustang GT (NFS Payback)
> Lucas Rivera's Chevrolet Camaro SS
> Ford Mustang Foxbody Starter
> Nissan Fairlady 240ZG Derelict
> BMW M3 GTR Most Wanted Edition
All Vol.2 Update Vehicles (Unlock by default, mp need to complete the vanilla challenge)
Vol.3 Update Vehicles
Vol.4 Update Vehicles
Vol.5 Update Vehicles

Story stage reward:
> Ana Rivera's Nissan 350Z
> Elainahan's BMW M4 Coupe '15 (Unbound Remix Test Custom Car)
> Subaru BRZ (NFS 2015 E3 Gameplay Demo)
> Morohoshi's Lamborghini Diablo SV
> McLaren P1 GTR James Hunt Edition
> Polestar 1 Hero Edition
> BMW M3 GTR Most Wanted Edition
> Lamborghini Aventador SV

High Heat Delivery:
> Zephyr's McLaren P1 (Complete Honor Amongst Thieves High Heat Delivery)
> Nissan Skyline GT-R R34 Heat Remix Edition  (Complete The Heat Remix High Heat Delivery)
> Nissan Silvia K's Reveal Trailer Edition (Complete Bait and Switch High Heat Delivery)
> Dodge Charger SRT8 `(Complete Grand Theft 5-0 High Heat Delivery)

Calendar Reward:
> Nissan Skyline GT-R R32 "Rydells Rides" (Week 2, Thursday Night)
> Mazda RX-7 "Rydells Rides" (Week 2, Thursday Night)

> Jaguar F-TYPE R Coupe "Build of the Week #7" (Week 3, Wednesday Night)
> Nissan Skyline GT-R R34 "Payback" (Week 3, Wednesday Night)
> Porsche 911 Carrera S "Lucas Rivera" (Week 3, Wednesday Night)
> Dodge Challenger SRT8 "House Enforcer"  (Week 3, Wednesday Night)
> Aston Martin Vulcan "Build of the Week #5" (Week 3, Thursday Night)
> Aston Martin DB11 Volante "Black Market" (Week 3, Firday Night)
> Ford Crown Victoria "Save Ferris" (Week 3, Firday Night)

> Koenigsegg Regera (Week 4, Wednesday Night)
> Lotus Exige S "Bait and Switch" (Week 4, Wednesday Night)
> McLaren F1 "Black Market" (Week 4, Thursday Night)

Reward for completion of Takeover practice race (Tuesday day of the first week): 
> Infiniti Q60S "Barracuda"

If you have already completed the story, most of these cars will be unlocked