I'm assuming then that:

Using a Higher-Level Spell Slot. All the damage increases by 1d6 for each spell slot level above 1.

... means that both the initial damage and the per round damage increase. Thus, a 5th level smite would inflict 5d6 fire damage on a hit, then potentially every round for the next minute.

Quote from DallonChevalier >>

I'm assuming then that:

Using a Higher-Level Spell Slot. All the damage increases by 1d6 for each spell slot level above 1.

 

... means that both the initial damage and the per round damage increase. Thus, a 5th level smite would inflict 5d6 fire damage on a hit, then potentially every round for the next minute.

 true and if im reading it right the enemy doesnt make their save until after they have at least taken the "burning" damage once from the effect. so at 5th level cast youd at least get 5d6 + 5d6 then its likly the monster would make its save bc monsters tend to have very high con. its like an upcasted fireball but only single target tho can do more but isnt very likely (unless monsters are getting changes to their con stats in the new MM) 

Wow, 2nd round of damage guaranteed, the effect can only end with a successful saving throw, and upcasting affects the damage over time. Great smite option, outdamages Divine Smite at 1st level if they fail the first save, and beats 2nd if they save first try (granted, fire damage is more likely to be resisted).

Factoring in multiclassing, a Sorcerer X/Paladin 1-6 will still eventually get 9th level slots, pushing this to 18d6 before any save is made. You'd need a critical hit to bring it up to 27d6 so it can compete with the likes of Meteor Swarm, but the fact that it can go this far as a base level 1 spell is quite the achievement!