I also recommend that same IC and it's precisely why I have some in my parts drawers.
I doubt you get better per pin current with either HC or HCT. HCT was intended to interface the CMOS parts with Legacy TTL (Meaning 5V logic 5V supply). HCT would not imply higher current capacity per pin.
My Philips datasheet says MAX Package current (Vcc to GND) -is 70mA... so your number has to be wrong. If each pin were driving a 10mA load, you will exceed recommended current for the 74HCT595 device.