Good afternoon. Please give me advices where I can send probes to find the continuation of my two-ellipses galaxy. I highlighted with red circles where I sent lots of probes but to no avail.

What does the wormhole technology do? please, can you explain in detail ? in my case, are the wormholes going to be the only way to traverse the galaxy?

There are constellations (like "Pisces" in your picture) where you can traverse on regular star paths, but eventually you will have to research the wormhole tech to discover paths to OTHER constellations. After getting the wormhole tech you should see the wavy paths that you could not see before. I believe the name of the tech is "Autonomous Materials", but I cant remember if any factions have it under a different name. It should be in the 3rd tier of the bottom (science) tech tree if you cant find it.