Write to Galaxies
Treat galaxies like CD drives to send directional laser updates of our civilization, and survive in the minds of the distant future civilizations.
Given that METI (MEssaging to Extraterrestrial Intelligence) is generally thought to be risky (e.g., "the fear of annihilation by an advanced civilization may not be limited to Earth, and it may help explain the silence of the cosmos"), it may be unlikely to have those civilizations try to signal to all the stars in the Milky Way, and upon picking up such signals, they would likely keep silence to keep strategic advantages.
So, statistically and from risk analysis perspective, we'd probably have better odds at discovering aliens by searching for them in other galaxies. Imagine that we do intercept an informative signal of an intelligent civilization somewhere out there in the Andromeda galaxy. We'd know something very special, down to a particular star system, and we'd be guessing, whether it still exists, because it happened 2.5 million years ago. We'd probably even feel safe to send a signal back, and continue broadcasting, especially if the signal that we picked up were to be periodic like blog updates about that civilization.
If the signals from aliens in another galaxy were periodic and lasting, with new bits of information every time, it would certainly be something that stays breaking the news here in the Solar system. Inspired by that, a project from our side could be to set up powerful lasers, pointed to the nearby galaxies, and broadcasting our status like that not to our local, but to the future civilizations that don't yet exist in other galaxies.
A single laser could send out bursts of packages to cover entire Andromeda, treating Andromeda like a CD drive, and using same techniques to write to CD drive with a laser (just more bits per dot), over a period of say, a month, and repeat that. This way, all the civilizations in the Andromeda galaxy, in the line of sight to the Milky Way could get monthly updates. That would certainly increase the odds of them discovering us, and if other civilizations in other galaxies had the same way of thinking, and pointed their lasers to the Milky Way, perhaps we just need more powerful telescopes to pick that up: the intergalactic chatter might actually be a real phenomenon, that happens, cause: over very large distances, even the potential enemies have enough time to become friends.
It's amazing to think, that "if you take our most powerful lasers, and our largest mirrors, and pair them together, for a fraction of a second, they can produce light that outshines the Sun by factors of a thousand at more than a thousand light-years away." -- so the method that this particular way implies, is that of using such extremely directional lasers, to make them see a new star appearing and flickering in the Milky Way, so that, with sufficiently large telescopes they could read our "blog posts" from.