As @DanP says, you can definitely do this. Some people find it necessary to recharge the transmitter, but I never did. For one thing, I couldn’t get all that tape off without accidentally yanking the whole sensor out, and for another I seemed to always have plenty of battery power left for the whole second session. Not sure why battery usage varies between people but apparently mine was relatively low. System seems to let you keep going until it gets down to about 30% and then it really demands you charge the thing, which was the limit for me, right around the 14 day mark.
I never used Enlites but I gather it’s pretty much the same procedure. Stop the sensor; disconnect the transmitter from the sensor–I found a quarter inch separation sufficient–wait 15 seconds or so; then push the connector back in until it clicks, watch for the green flashing light, then turn the sensor back on and select new session from the sensor connection menu.