Partially fill a 1 gal. container with well-draining potting soil. To make it more acidic, add some eggshells and some powdered quicklime.
Look under the parent oak leaf hydrangea. Find sprouts that are 2- to 3-inches tall.
Dig up the sprout or shoot with a trowel. Make sure you leave as much as the original soil as possible around the root ball.
Place the sprout or shoot into a 1 gal. container that you have partially filled with potting soil. Finish filling with potting soil, leaving about an inch of space between the soil level and the rim of the pot.
Water your shoot thoroughly. Keep the hydrangea watered so soil is moist, but not soggy. They will rot if their roots are kept constantly wet.
Find a place that offers some morning sun. Keep your new plant out of direct sunlight. In a week or two, you can move it to a sunnier location.
Continue to water and care for the hydrangea for a period of 60 days. After that, you can plant the hydrangea in the yard. Plant in a place that receives filtered sun, but do not plant under a tree. When you plant this in the yard add some eggshells and some powdered quicklime to the soil as hydrangeas like a more acidic soil.