I export on Octopus Intelligent Go @ £0.15 per kWH.
If you really wanted full control you'd go Octopus Agile.
And as the other guy said, you don't have to wait for overnight either to get cheap energy for your car, I can literally plug in any time and as long as there is excess capacity on the grid it'll dump it in the car @ £0.07 per kWh. You're not limited on charging at a certain time either, if you need to charge the car now you can charge it straight away, you just pay for it.
With Agile you could control it on a 30 minute basis - or set up automation with IFTTT via Home Assistant. You actually have less control now than you think compared to others on the Intelligent or Agile tariffs, with standard Octopus Go you're locked into cheap electric only between 00:30 and 05:30, and outside that good luck, especially on a car with a 100kWh battery with a 7kWh home charger.
With Intelligent Go the chances of you charging from 0% to 100% are actually way more realistic in a 24 hour period vs. what you have now. Yeah I know it's rare that most people do it, but I do sometimes with things like airport trips.
Might be worth running the numbers on upgrading your charger vs. the cost of a couple of years at the lower rate - you might be surprised how quickly the new charger will pay for itself, it was the calculation I did vs. 3-pin plug charging when I got my XPOWER.
I can still control how much power goes to the grid, to my eddi, or zappi. As you can imagine with an XPOWER you charge quite a bit, which will change a bit when my IM5 LR arrives, but then again it's got a much larger battery so it'll be a case of charging for longer but much less often.