You can attempt to maximise your chances by doing a couple of things.
When you arrive in the system and use your scanner you may only, as per the OP, find the star you've arrived at. even though the system map indicates there are two, three, or more other stars. A companion star may be so far away that your scanner can't pick it up.
Once you've used your scanner the orbit line of your current star should be apparent. If it's not, then that star isn't orbiting around anything as it's a lone star and there are no other major bodies in the system.
When you have the orbit line, orient your ship so you can pitch along that line and see if there are any stars on it. Let's assume you see a bright star on the orbit line and that system map indicates there is a single companion star. That companion star may be the one you're looking at but it may not. It also wont be actually "connected" to the orbit line you have on the screen because that orbit line is only the orbit of your current star around the barycentre of the star system. But, critically, it does show you what plane the two bodies share.
Now reorient your ship so the orbital plane is horizontal across the screen. Pitch down or up so the star in question is at the limit of your visibility and throttle up.
What you're doing is accelerating out of the orbital plane of the binary star system. This maximises your chance of observing parallax movement against the background of other stars.
You're looking for obvious parallax movement of the star in question against the background. So keep an eye on your target star. Your actual orientation doesn't really matter but it's easier to understand what you're doing if you keep to the planes and other cardinal angles.
If you do see the parallax movement you can be sure you've got the other star and can reorient your ship and go straight at it.
You may not see any parallax movement. You may get to 1000c or more and still not see it. The companion star may be that far away. On my trip to Sol I found a couple of systems where this happened.
In these circumstances the only thing to do is look at the system map and see if you can gauge the characteristics of the companion star and then look along the orbital plane for something that looks a bit like it. Then head straight for it at high speed.
For example, one of the systems I went through had a big yellow star, a small yellow star, and a purple brown/red dwarf. The purple one was miiiiiiiles away. I found it on the orbital plane and went for it.
Took half an hour of SC before I could detect any parallax at all.