Your math is fine, but the speed of the fastest probe we have now is no indication of what is possible. 238 km/s is less than a thousandth of the speed of light.
Laser- or maser-powered light sails have been modeled as attaining perhaps 30% of the speed of light on interstellar missions. The Breakthrough Starshot project is hoping to use this technology to get to Alpha Centauri sometime later in this century. Many technologies would need to be developed to make this possible, but the physics of it are well understood and sails propelled by sunlight have already been flown.
An earlier project, Project Longshot, looked at doing the same thing using pulsed fusion propulsion. Controlled fusion would need to be developed in order for this to work, so it was more speculative, but again it relied on physical phenomena we know a lot about, though we can't reproduce them yet.
And there are several other options for such things. The issue is that it would take tremendous infrastructure and technology development to launch them. But we have good ideas as to how it might be done.
Of course, if it is going to take 30 million years for your craft to get outside the Local Group, you have other things to worry about besides how fast it can go.