I did the same route in the opposite direction about that time last year. However, this year has a precipitation profile so different than last that my observations would be totally irrelevant. For example, the only snow I saw on a trail was at the Old Army trail where it connects to the New Army trail (I took New Army, but wanted to check the Old trail's conditions so I took a peak from the top).
One way to monitor the snow levels along that route is to check the Crabtree snow sensor at the link below. You will most likely hike by this sensor (if taking the PCT and then JMT) from Horseshoe Meadow. As of today, Crabtree Meadow has 39 inches of snow (8.7 in of water equivalent). Of course, this is just one point at 10,700 ft, but the data can help you understand the melt rate in the region as a whole.
I hope this helps. http://cdec.water.ca.gov/dynamicapp/QueryF?s=cbt