while driving the car and coming to a stop and restart: as the car stops and not necessarily to a complete hard stop, the transmission hesitate to go completely into first gear. If I press the car then the transmission downshifts hard into first. All other shifting is normal.

 

When were the trans fluid, filter, etc last changed?

 

Probably about 3 years ago.

 

It may be low or if you do lots of miles due changing again.

 

Sounds like the lurch, fairly common fault but can be hard to cure, you can search here or check out......
Jaguar S-Type Ownership Experience - The ZF LURCH Fault