Other than refinement, upgrade the charm equipment too, Head Charm upgrade will boost the initiative stats.

1) Purify Equipment, try to get yellow stats initiative first then locked if u planning to yellow stats the others

2)Eight Inner Gates, requires alot coins and spamming. Redeem the yellow Quick Chakra that gives initiative

3)Five Nature Chakra, Position 1 gives initiative increase for each stage upgrade

4)Assist link, level 4x4 sets to lvl 9

5)Mt.Myoboko, Ninja Assist and Secret Scroll, they work basically the same. Each upgrade gives stats to all pos of ninjas. So more upgrade, more stats increase.

6)Treasure tools, focus 1 or 2 rope first

7) Ninja Breakthrough Attribute, this also contributes to overall stats. Some nin bt gives initiative. If not mistaken each ninja max initiative increase cap when they reach break level 12.