When to push deer?

I've recently started trying to push deer in and it works quite well, but I'm a bit unsure about the timing. Usually, I start pushing when I've found my two boars and at least 2 sheep. Mostly that's exactly the moment when I'm about to get the first boar or I already have it. Is that the right timing? I'm wondering because boar decays faster than deer, so generally you don't want to move vills from boar to get deer, right? Is it a good idea to delay the boar or get it earlier, or just push the deer later?

Any ideas or suggestions would be much appreciated