It appears to me that Global Poker has held these tournaments around the following dates (correct me if I'm wrong)...

Eagle Oct 2017
Grizzly Feb 2018
Rattlesnake May 2018
Eagle II Oct 2018
Grizzley II Feb 2019

Does anyone know if there will be a Rattlesnake II coming soon, or when the next Tournament Cup will be?

Any missing that I didn't list?

Yes..you are missing the 1st Series from June 2017..here's an announcement from 5-31-17...
$1,000,000 in Cash Prizes Guaranteed - Online World Challenge
Global Poker is proud to announce its very own Online World Challenge. During the course of 8 weeks, we will be giving out a total of $1,000,000 $weeps Cash in guarantees and freerolls!

I began playing on Global a few weeks after this and it was just finishing up. Aside from Global doing away with a few good ideas like the "Weekly Gold Coin Challenge" that required a little GC play to unlock a SC Leaderboard challenge you are left with the series you mentioned.
Bring back the Leaderboards Global!! ��

In my experience they are a little slow on the draw in announcing series. There may well be a Rattlesnake II in May but I wouldn't expect it to be announced quite yet based on past experience.

Cool. Was that tournament series named after an animal?

No..it says in the message..it was called Online World Challenge. The irony of course is that how can it be a World Challenge if only the US and Canada can play on global.