Viruses don’t often switch species: there are many barriers. In order to succeed, viruses must interact correctly with hundreds of proteins – ones they make use of, and immunity proteins that might destroy them. Not easy, and so less than one in a thousand bird and mammal viruses seem capable in infecting humans. This can’t be a product of gradual evolution: the viruses has to be capable of infecting human cells from day 1. Virologists often attempt to grow important human pathogens in laboratory animals, and usually it doesn’t work. Second, the virus, even at at the very beginning, has to have an R0 > 1, else it will quickly burn out. So, even at the beginning, it has to be able to infect humans and be transmitted at least moderately efficiently to other humans.
That doesn’t mean that there isn’t room for improvement. Once our new virus starts spreading, mutations that further its spread will be favored by natural selection. That selection is extremely rapid. Virologists make practical use of that rapid selection. For example, they wanted a mouse model of SARS. They did manage to get SARS to infect mice, but it didn’t cause as serious a disease as in humans. They passaged the virus 25 times: infected a mouse, injected a sample from that infected mouse into another mouse, and so on, resulting in a strain that was highly lethal to mice. Natural selection in the course of the infection of a single mouse led to a better-adapted strain of coronavirus: repeated 25 times, it transformed the virus from not too serious to highly lethal.
In an analogous experiment, researchers took a subtype of an avian influenza virus (one not known to infect humans) and passaged it in ferrets. Originally, it could infect ferrets but was not transmitted noticeably to other ferrets: but after 10 passages. it could – without losing virulence.
“At this point, the virus population will experience selection for variants with increased capacity to spread through the human population.”
Covid-19 has been not in humans very long and is probably changing fairly rapidly.