Heat five 12-ounce jars in simmering water until ready for use. Wash lids and rings in warm soapy water.
Fill a large pot with water; bring to a boil. Add peaches; cook until skin loosens, about 20 seconds. Drain and cool until easily handled. Peel, pit, and cut the peaches into chunks.
Measure out 5 cups of peach chunks into the pot. Mash with a potato masher. Add sugar, peach gelatin, lemon juice, and almond extract; mix well. Bring to a rolling boil and cook jam for 1 minute. Remove from heat.
Pack jam into hot jars, filling to within 1/4 inch of the top. Wipe rims with a clean, damp cloth. Top with lids and screw on rings.
Place a rack in the bottom of a large stockpot and fill halfway with water. Bring to a boil and lower in jars using a holder, placing them 2 inches apart. Pour in more boiling water to cover the jars by at least 1 inch. Bring the water to a rolling boil, cover the pot, and process for 10 minutes.
Remove the jars from the stockpot and place onto a cloth-covered or wood surface, several inches apart, until cool, about 12 hours. Press the top of each lid with a finger, ensuring that lid does not move up or down and seal is tight.