Congratulations!
Now, later this week the August shop will be available, if it's not already there at the ridicu-low price. On August 1 or shortly thereafter, it will have its first increase, which should be significantly more than the 26 cents or 12 cents that started this thread, maybe "starting" at $5.80-$6.25.
The frequency of increase will not change, so "rising more quickly" in this case really means rising more each time.
I don't know what the re-shop timeframe is for shoppers for this client, but if it's thirty days, and since you seem to only have one competitor shopper for this location, you could wait until August 28 to take the shop for yourself, at which point it should be a lot more than $12.50 since it will be starting higher.
However, as soon as it passes $12.50, it may be enough to lure a third shopper into your area - your cheapest competition was $12.50, but there will be others. The next cheapest competitor may have been waiting for it to hit $13 or $25 or... who knows?
It's pretty safe, though, to say that you can wait until it approaches $12.50 to take it in August. Since you seem to have been diligent in checking the price regularly, you could probably even be safe in taking it at the first increase beyond $12.50 if you can be certain to catch it before anyone else does.
The only radical uncertainty that could scuttle your strategy would be for a newly registered shopper to pop up in your neighborhood and change the calculus by taking that as one of their first shops, not knowing what the (newly discovered) going price is.
Have fun getting to know the Cori game and learning how to win at this particular location! I have a GS shop that is basically "my" shop, going two months out of three to me for $20-28, but since it's a 45-day re-shop, I have to let it go to someone else every third month for $30+ because that's what it take to lure someone so far out into the sticks (but that boosts the price for me again for the next month.)