Bluetooth devices should auto pair once both devices are turned on and within range, unless you have the jambox also paired with another device that may be on. Most speakers do not dynamically change devices like a car does with multiple phones, so if you are pairing with another device, you will probably have to re-pair when you want to switch to another Phone.
Any how, I do not believe the issue is with you iPhone in the end since all of your other items are working properly. you can tap the 'i' for the Jambox in Settings > Bluetooth and delete the device and repair it fresh if you like to prevent future issues. After successfully pairing and playing some music, you should (normally) be able to turn off your iPhone and/or jambox and turn them back on and within a few seconds be auto connected. If not, contact the manufacturer of the jambox.
I hope this helps.