The last trail report I saw was in early May when there was too much snow to locate the spring, but I can say that Jackstraw Springs normally runs well into the fall every year. It can get down to a trickle later in the summer, but with patience you can still draw water from there.
If no one else responds before you plan to go, you may want to carry a little extra just to be sure, but I am confident that you will find the spring flowing up there.
Also, if you plan to continue to the ridge or the peaks beyond, you can also stop in at High Meadow Springs camp. The water was reported to be abundant there only a week or two ago. That spring also normally runs into the fall but can be more difficult to locate. Check the maps for the locations of the spring/creeks to help locate the water at these camps.