The order matters. Here is the right sequence, the wait between, and why most hot tub owners rarely need algaecide in the first place.
Yes, but in the right order and with a wait between. Shock first, run the jets for at least an hour, then add algaecide. Reversing the order or stacking them inside the same hour wastes both products and can cause foaming.
Most algaecides are quaternary ammonium compounds (quats). Chlorine attacks quats just as readily as it attacks algae cell walls. If both are at full concentration at the same time, the chlorine consumes the algaecide first and the algae get a free pass.
Shock goes in first because it is the bigger oxidising hit. Once it has dropped back into the bathing band over an hour or two, the algaecide can do its job without being immediately destroyed.
Hot tubs almost never need it. The temperature and the constant sanitiser presence make algae a non-issue. If you are seeing green, the cause is almost always insufficient sanitiser or biofilm rather than true algae.
Swim spas need it occasionally because of the larger volume and the lower running temperature.
Garden hot tubs that have been left without sanitiser for weeks may need a one-off algaecide treatment alongside a heavy shock to clear the slime, but that is a rescue operation, not a routine.
Shock with chlorine to 10 ppm, jets on, cover off. Wait two hours.
Test free chlorine. Once it is below 5 ppm, dose algaecide per the bottle.
Leave the jets running for another hour, then cover and leave overnight. Test the next morning before any bathing.
Quat-based algaecides will foam if dosed at the upper end of the bottle range. A small amount of foam is normal for a few hours.
Polyquat (polymeric quaternary ammonium) is much less foamy and is what most pros use in hot tubs when they need to use one at all.