Hi Irsida,
Greeting from Peggy, Top China Travel.
Yes, you need to pay another card to enter Tianmen mountain.
The 248 yuan entrance fee is only used for Wulingyuan Scenic Area. Wulingyuan Scenic Area covers four parts including Tianzi Mountain, Yuanjiajie Scenic Area, Huangshi Village, Golden Whip Stream.
And Tianmen Mountain is another attraction which is not included in Wulingyuan Scenic Area. So the 248 yuan entrance card could not be used when visit Tianmen Mountain.
And the validity of the card is 3 days, that means you can visit above four parts within 3 days, not including Tianmen Mountain.
Peggy
Top China Travel