Prepare the fruit. Rinse the fresh cranberries and strawberries thoroughly under cold running water. Remove the strawberry hulls and cut the berries into pieces.
Peel the orange and chop the individual segments. Finely grate the zest from the orange.
Place the cranberries, strawberries, orange pieces, and orange zest in a large cooking pot. Pour in 1/4 cup water or orange juice to provide enough liquid for the fruit to begin cooking.
Incorporate the sugar and cinnamon. Add the granulated sugar and cinnamon stick to the prepared fruit mixture. Stir thoroughly to distribute the ingredients.
Set the pot over medium heat and continue heating, stirring from time to time, until the sugar has dissolved completely.
Cook the jam. Increase the heat and bring the fruit mixture to a boil. Once boiling, lower the temperature sufficiently to maintain a gentle simmer.
Cook for approximately 10–15 minutes, stirring occasionally so the mixture does not stick to the bottom of the pot. As it cooks, the fruit will soften and break down while the liquid gradually thickens.
Check the consistency by placing a small spoonful of jam onto a chilled plate. The jam is ready when it thickens sufficiently to hold its shape.
Alternatively, check the temperature with an instant-read thermometer. At sea level, cook the jam to 220°F. For higher elevations, reduce the target temperature by 1°F for every 500 feet above sea level.
Fill and process the jars. Once the jam reaches the desired consistency, take the pot off the heat.
Carefully ladle the hot jam into sterilized canning jars, allowing 1/4 inch of headspace at the top of each jar. Wipe the rims clean and secure each jar with a two-piece canning lid.
For shelf-stable storage, place the filled jars in a boiling-water bath canner and process for 10 minutes. At elevations above 6,000 feet, process the jars for 15 minutes.
For refrigerator jam, transfer the finished preserve into jars, allow them to cool, and store them in the refrigerator instead of processing them in a water-bath canner.
Cool and store. Leave the jars undisturbed at room temperature until completely cool. After cooling, inspect each jar to confirm that a proper seal has formed.
Keep correctly sealed jars in a cool, dark storage area for up to 12–18 months. Refrigerate the jam after opening.