As the popular say goes, everyday in the hunting fields is a good day, but there are indeed a couple objective criteria to integrate in the decision making process.
Season
South Africa is essentially 6 months apart from the US when it comes to seasons. We are high in the northern hemisphere, they are low in the southern hemisphere. Our coldest months are their hottest months, and vice-versa. What this means in practical terms is that if you are familiar with the American West (Arizona, Colorado, Utah, Nevada, etc.) you will find a lot of similarities with South Africa ... offset by 6 months.
June/July/August in Eastern Cape will be very similar, for example, to December/January/February in Arizona, Nevada, etc. - excluding the highest elevations which South Africa generally does not have, compared to the US: I am talking about thousands of square miles above 10,000 ft for example. You will have occasional white frost at night in middle elevations with shirt-sleeve weather during sunny days, and fleece or puffer vest appreciated on cool mornings and evenings.
Conversely, December/January/February in southern Africa can be painfully hot (and wet) and game activity may be reduced to early and late hours only, with game (and hunters) panting in the shade during the middle of the day, which reduces the hunting opportunities.
Location / elevation
South Africa is again similar to the Western US: many of the best hunting locations are located at fairly high elevation in high plateaus or mountains. The Waterberg Mountains of Limpopo Province in the North, for example, have an average elevation of 2,000 ft and go as high as 6,500 ft; the Stormberg Mountains culminate at over 11,000 ft in the Eastern Cape Province in the South; and the Free State, in between, is located on the Highveld plateau rising to elevations of 6,000 ft in the East and sloping to 4,000 ft in the West.
Each general area will have much diversity with warm valleys and cold mountains, so you really want to know exactly where you will go as, like in Arizona, Utah, Nevada, etc. 100 miles can make a dramatic difference in relief, elevation, vegetation, temperature, ecosystem, game species and density, etc.
Weather
Combined latitude (seasons) and altitude (elevation), as well as ocean proximity or lack thereof, drive the weather in South Africa.
I personally use weatherspark.com to get a month by month Climate Summary, Average High and Low Temperature, Cloud cover, Precipitations, Humidity, Wind, etc. of the closest town to the concession I will be hunting. It has proved remarkably helpful in selecting when to go and what to bring. Here is an example for Queenstown, Eastern Cape:
https://weatherspark.com/y/92839/Average-Weather-in-Queenstown-South-Africa-Year-Round.
November to March will typically be the rainy months, with daily chance of precipitations going as high as 40%
December to February will typically be the hottest month, with daily temperature approaching the low 90F.
Game rut season
Many species of antelopes are in rut in the April/May time frame, South Africa's fall. Like with deer or elk rut in the US, the dominant males are prominently active and this creates many stalking opportunities.
Peak season
Many folks decide to go in the March to June time frame, with April/May being often the peak season. These are great months in term of autumn weather (dry, although April can still be occasionally wet, and warm, but not too hot) and in term of game activity (rut).
Consequently, this is the busiest season and hunting camps and lodges may be crowded.
From a financial perspective, prices may be more attractive at the low season in the November to February time frame in places that are hunting all year long.
Best time to hunt?
It is all a matter of personal preference.
Some prefer the warm days of autumn (March/May) with rut activity, and they do not mind a higher hunting pressure.
Others prefer the cool days of winter (June/August) and lower hunting pressure.
Others yet prefer the heat of summer (December to February) and do not mind an occasional shower for a different African experience.
I personally prefer the African winter months (July and August), because they are more like late Autumn weather in America and Europe, crisp in the morning and gently warm during the day, not too hot for either human and animals, with reasonable game activity most of the day, lower hunting pressure, and fewer hunters in camp.
Bottom line
As many said, above here, discuss with your outfitter and state candidly your likes and dislikes and the ranking of your priorities. There are many options...
I hope this helps in giving you the reasons behind the choices