Rolling for Hit Points

In our last campaigns when players were leveling up, I allowed them to take the average value of their hit dice if they rolled below. 

For this campaign, I would like to rework this rule. Instead of taking the average, players will take the value of the dice plus their Constitution modifier. If a player rolls a 1, they can choose to reroll and take the new value. They can continue to reroll if they continue getting 1s.