If using solar energy to heat steam reduces the per-unit cost of extracting each possible quantity of oil, what happens to the average total cost curve?

The average total cost curve would shift downwards because of this action.
The Average Total Cost is calculated by dividing the total cost by each unit produced, and the total cost is the sum of the fixed costs and variable costs.
Using solar energy reduces the cost per unit of extracting oil, so you are reducing the variable costs, while the fixed costs remain the same.
For instance, if the fixed costs are $100, and it costs $2 to extract each unit of oil, your average total costs for 100, 200, and 300 units of oil produced would be $3, $2.5, and $2.33. If, however, using solar energy reduced the per-unit cost to $1, the average total costs for 100, 200, and 300 would all drop to $2, $1.5, and $1.33, respectively.
If the solar energy had a higher fixed cost, it would still do the same for the Average total cost curve—for example: Fixed cost = $150, Per-unit cost = $1, average total costs for 100, 200, and 300 = $2.5, $1.75, and $1.5.