First, please note that in technical economic terms, you can never increase supply by increasing the price of a good. A price increase can cause a change in the quantity supplied, but it cannot cause a change in supply. This is an important distinction.
In order to increase supply, a producer must create a situation in which they will be willing and able to produce more of the good at the same selling price. Let us say that I have been producing 100 t-shirts a day at a price of $5 each. In order to increase supply, I must be willing to produce more than 100 shirts each day while they continue to sell for $5 each.
The most common way to do this is through reducing my costs. If I can reduce my costs, I will make more profit on each shirt. If I can make more profit, I will be willing to produce more shirts. Therefore, I might decrease my workers’ wages. That will lower my costs. However, I might lose workers that way. So another thing to do would be to increase productivity. I could buy better sewing machines for my workers or redesign their work spaces so that they could make each shirt in a shorter period of time. Both of these would reduce the time it took to make each shirt and, thereby, the price I have to pay in labor costs.
So, the best way for a producer to increase supply is to reduce costs.