You don't need to take the brifters off to do this.
Simply "lift the skirts" of the rubber hoods and you'll see how the current tape was wrapped. Take some photos before unwrapping.
Personally I like a bit more padding from the corners to thge hoods so I will over-wrap that area, or perhaps wrap some old tape there before covering it with new bartape for two layers. Extra important on thin 80's style bars.
If you're using new tape it will probably be longer than you need so expect to trim it when you feel happy with the wrap job.
My technique is to start at the bottom and always bring the new tape "up the outside and down the inside" which means going around the brifter to reverse the flow ready for the tops.
That bartape looks kinda cheap which contributes to its early failure. Try and use something better, definitely not $2 foam tape. Avoid light colours, they show the dirt easier. Basic black suits most things, but depending on the rest of your bike and its colour scheme you might choose a matching or a contrasting colour.
I prefer red for the lower part and black bartape on the tops and hoods.
You could also practice by removing the old bartape and re-fitting it before you stick-down any new tape. Expect to unwind it and have a couple of attempts, its all a learning experience.
Try re-fitting tape as it was, and then swap sides so the "set" is unhelpful. You might find that the scungy tape ends up elsewhere and you get another window of service out of the same tape. Maybe.