The best information of the JMT this time of the year are the PCT hikers as they are the ones with first hand knowledge. This is the current snow conditions (as reported by hikers) from the website PCTWater.com that handles a lot of first hand PCT info like water conditions along the trail. Scroll down to mile 799 to see info on the Woods Bridge being intact.
PCT 2009 / AT (ME->MA) / CDT SOBO in 2023
Tahoe Rim Trail 2014 / Condor Trail 2016 / John Muir Trail 2017
Uinta Highline Trail 2021 / High Sierra Trail 2022 / Wonderland Trail 2022