With the updated mod, you can skip the whole "escape the Hive" part and start right at a city already. But if you prefer to try to escape yourself, here's a guide.

There are now three different starts for escaping the Hive; the Village, the Armory, and the Queen's Hive. Since the Village was the original start, I'll go through that first.

As soon as you start, immediately run indoors to grab supplies. Look for medical supplies first, which you absolutely WILL need. Start looking in the barrels and boxes before snatching anything lying about outside of them, because as soon as you snatch something, you're a thief, and then the workers in addition to the soldiers will be after you too. And the workers are really hard to avoid indoors with their long staves! It makes stealing anything else a challenge. I paused frequently to plan out my next dash while trying to dodge the soldiers during the search. You will likely have to search multiple buildings to find a medkit, but I pretty regularly managed to find at least one in the four buildings.

Now, once you have the medkit, dash to a "worker" building (a building that has workers spawn inside) if you're not in one already. In there let the guards knock you out. There's almost no chance of you escaping the guard, but if you're lucky, you'll only be injured just enough to briefly lose consciousness and get the guards off your back.
I've found that the soldiers don't often enter "worker" buildings when not chasing you, and if you find a secluded corner and stay in stealth mode, you can go unnoticed until nightfall. During this time, build up your thieving skill by repeatedly snatching up and then dropping to snatch up again, whatever's lying around on the floor. Do NOT take anything in a barrel or box yet, as you will likely get caught and then get another potentially fatal beating.

Once it's dark, take what you want for the journey, including from the barrels if your theft skill is enough to have good chance of stealing from them without getting caught. I suggest not taking heavy armor or anything else that will hamper your stealth, though. Then make your escape.
While moving about, pause frequently to scan for patrols. I carefully planned each move in short darts from one hiding spot to another, so I was never far from a hiding place when a patrol suddenly appeared. Keep moving if you can; in Hiver territory there is practically no hiding spot that patrols won't eventually find. Eventually, the patrols will thin out, so you only have to watch out for beak things and gorillos.

The Queen's Hive start is very similar to the Village start, except the soldiers will freak out about you threatening the Queen if you enter the large, non-hive building. You can use the same essential tactics as the Village start.

In the Armory start, you'll start near an armory building with no soldiers immediately about. Depending on which armory it is, there may be hive buildings nearby or stuff lying about on the ground. If there are hive buildings, don't enter them, because that's where the soldiers are. Get out of there as fast as possible, before the soldiers come out and see you. You can quickly grab a few items lying on the ground, though it won't be anything immediately useful for your journey, like food or medkits, except maybe the tools if you get caught and bound to a pole.

I went north, straight to Lagoon Flats. It is a pretty good starting city, if you can make it there; the frequent attacks on the gates from animals means you can collect plenty of skins and meat to sell, and it being a Tech hunter town means you can buy some good weapons and armor once you've saved up enough cats. It also gives you a chance to practice fighting on wounded animals.
I save-scummed like a cheating cheater to make it out, though I tried as hard as possible to do it as little as possible. No matter what you do, this is going to be a really hard start that you will have to restart many times before your pink antman makes it out of Southern Hive territory.