The new way to get time extensions is still being tested, but here is what I have found so far

Also, as far as I can tell all the zones have special names for the times. Bastok is images iirc, xarc it's prototype X, Jeuno it's statues. The non-te stones are replicas.

Getting certain TEs seems to unlock the ability to !! stagger different ways, increasing drop rates

Description for the TE Key Items from wiki:

Azure granules of time
Thought to have once flowed within
the prismatic hourglass, these
granules of sand cast a deep blue
light.

Amber granules of time
Thought to have once flowed within
the prismatic hourglass, these
granules of sand cast a yellowish
light.

Alabaster granules of time
Thought to have once flowed within
the prismatic hourglass, these
granules of sand cast a pure white
light.

Obsidian granules of time
Thought to have once flowed within
the prismatic hourglass, these
granules of sand cast an opaque,
black light.

Crimson granules of time
Thought to have once flowed within
the prismatic hourglass, these
granules of sand cast a dark red
light.

Judging by the descriptions it sounds like getting TE figures in there somehow.

0 TE - No Proc
1 TE - No Proc
2 TE - No Proc
3 TE - Blue Proc - Difficult to proc - Nothing
4 TE - Grellow Proc - Medium to proc - Terror/Stops 2HR/able to reuse
5 TE - Red Proc - Easy to proc - Terror/Stops 2HR/unable to use

As far as the job procs go
JA Procs on Rng Nin Bst Thf Mnk
WS Procs on Pld Drk Drg Sam War
Magic Procs on Whm Blm Smn Brd Rdm