Solved

S9, S6 etc.

It's not driver reputation, it's manager / team reputation. There are two numbers, one is your current reputation which will normally change after each race. The other is the peak reputation which is the highest rep your team has ever had and the S? in brackets after the number is the season in which you achieved it.

Reputation is reset at the start of every month, we are currently on season 10 (S10), your peak reputation will have been at the end of S9 (28th Feb) because you only started playing last month.

In Gregory's case it is not the reputation he had when the HoF was changed, he wasn't playing the game then.

Agreed, but I stand by the statement, it's the highest reputation ever achieved by the manager. 

If they've been playing since before July 1st 2020 then coincidentally it is most likely that this is their rep score on 30th June 2020.

Sorry Dome but I don't agree that your statement is more correct.

If a manager started playing before June 30th 2020 and for the first few races they were still finding their feet and learning the game it could easily be the case that on June 30th 2020 their rep score would be less than 5000.

In cases where rep is less than 5000 it is increased back to 5000 at the beginning of each month. So there will be exceptions to your statement.

In any case I wasn't discussing the merits or drawbacks of the current version of the HoF, I was just trying to answer Gregory's question about the "S" number, which is the season that you achieved your peak Rep score

Which, if you've been around long enough will most likely will be S1 LOL.

Aha, I now understand your point and of course you are correct.

The S number refers to your highest rep score on the final day of the month commencing 30th June 2020. 

Where:
30th June = S1
31st July 2020 = S2
31st August 2020 = S3 etc.

Does that work as a better explanation?

I suppose most importantly... Do you understand Gregory?