In a large bowl, combine peaches with the brown sugar, lemon juice and about half of the rum, stirring to mix. Cover and let stand at room temperature six hours or overnight.
Wash jars and lids in hot, soapy water. Sterilize jars for 10 minutes in a boiling-water bath, then leave in hot water until ready to fill. Prepare lids according to manufacturer's directions.
Pour the fruit mixture into a large saucepan or dutch oven. Bring to a boil over medium-high heat. Cover the pan, reduce heat and cook the mixture until the peach chunks begin to look translucent, 15 to 20 minutes; stir occasionally to prevent sticking. If the jam becomes too thick and threatens to scorch before the fruit is done, add 2 to 3 tablespoons of water. Add the granulated sugar, increase heat to medium-high and cook rapidly, stirring almost constantly, until a spoonful placed on a chilled saucer and refrigerated for a few minutes wrinkles instead of runs when the saucer is tilted. (Take jam off the heat while doing this. If using a candy thermometer, this should happen at about 220 degrees.) Add remaining rum and stir the jam (it will boil up when you add the rum) for 2 minutes over the heat.
Ladle boiling-hot jam into hot, prepared jars, leaving 1/4 inch headspace. Top with lids and process for 15 minutes in a boiling-water bath. Cool jars completely on a dish towel before labeling and storing.