In a big pot, combine the tomatoes, onions, tomato sauce, and jalapeños. Bring to a boil over medium heat, stirring constantly.
Reduce the heat and simmer gently for 5 minutes, stirring frequently to prevent sticking. Add the lime juice, zest of a lime, and Tabasco, then gradually stir in the sugar.
Heat the mixture over medium-low heat, stirring constantly, until the sugar is completely dissolved. Increase the heat to medium-high and bring to a full rolling boil, stirring constantly.
Stir in the pectin. Return the mixture to a full rolling boil, stirring constantly. Boil for 1 minute, stirring constantly.
Remove the pot from heat and skim off any foam. Let the jam cool in the pot for 5 minutes, stirring occasionally.
Spoon the jam into hot jars, leaving ¼-inch headspace. Remove any air bubbles. Wipe the jar rims and threads with a clean, damp paper towel. Apply hot lids and screw bands.
Process 4-ounce, 8-ounce, and pint jars in a water bath canner for 10 minutes. Remove from the water bath canner and let cool for 12 to 24 hours. Check the seals and remove the screw bands.
Stash jars in a cool, dry, dark place for up to 12 months.