The blast of water from the hose won't be strong enough to clear the blockage . The pressure from the tap is not strong enough . You should try the big bucket of water and throw in in to the toilet in one go .
The tap in question is the garden tap so the pressure is already good. It has already proved partially effective.
I should have added that the toilet in question is one of those 'modern' lo-flo ones, i.e. the water cistern is much smaller than on the original (1060s) bathroom in the house.
Not having a clue about plumbing, I am unclear how pouring water down the bowl adds to the force of water in the flush ? If there's a blockage which is manifest in the water level in the bowl, post flush, rising to almost the rim, will adding water to the flush not just increase the risk of an overflow ?