"Likely"/"most feasible" are going to be contentious, since many people make a lot of implicit assumptions about what advanced civilisations are like and what their goals would be. That said, given currently understood science, there are many options for getting to other galaxies far earlier.
As Stéphane pointed out, the escape velocity of the Milky Way is around 550 km/s - far higher than any current rockets and well above anything plausible for chemical rockets (since their exhaust velocity is on the order of km/s, the rocket equation requires a vast amount of reaction mass - for a 1 kg payload with a 4.4 km/s H-O rocket we need $\approx 10^{54}$ kg reaction mass). Using a VASIMR wiht 100 km/s gives a mass ratio of 244.69, and were we to use a photon rocket the mass ratio is just a few percent (but now we need some awesome energy source to power the laser or photons).
If you do not use rockets but launch using external acceleration things look even better. Breakthrough Starshot hopes to be able to launch very small probes at 20% of lightspeed within a few decades. That would easily get out of the galaxy, although they are aiming at nearby stars. I have a paper analysing a more advanced project where we use a Dyson sphere to power the probe launch, and we find that as far as physics goes, it looks possible to send probes over vast intergalactic distances - enough to settle everything within hundreds or thousands of megaparsec (the big uncertainty is dust densities, although one can honestly debate the best form of braking - now we think detachable laser projectors are the way to go). Getting anywhere within the local cluster would take a few million years of transit.
Even if one doesn't build a rocket one can leave the galaxy. The galaxy emits hypervelocity stars, likely due to binary encounters with the central supermassive black hole. They have velocities above galactic escape velocity. In another paper I sketch out an argument that one can nudge stellar orbits to direct streams of hypervelocity stars to move galaxies (I have updated calculations in greater detail for a book project, finding a $\Delta v$ of about 48 km/s for the Milky Way if we use all stars); while moving entire galaxies is very cumbersome one could certainly use the stars and their solar systems as spacecraft. The speed is not great, it will take hundreds of million years to get to Andromeda.
In short, we do not need to invoke hypothetical physics to get to other galaxies. But the most plausible schemes will take a long time by human standards and favour very small craft - this is more the realm of postbiological intelligence than astronauts.
Also, the clock is ticking: if the acceleration of the universe keeps on going, within a few hundred billion years galaxy clusters will start separating at an exponential rate and become isolated. There will not be any plausible way to travel between them and we will enter an era of isolation. Stars lost between them will become totally isolated, with nearly no chance of joining any of them. So we may want to build our Dyson spheres relatively soon.
