You have outlined the principal decomposition path of hypochlorite in the question. Chlorate in itself is rather stable under ambient conditions and would probably not decompose.
The greatest danger is if the bleach solution is no longer alkaline. If acidified, both hypochlorite and chloride as well as chlorate and chloride can comproportionate to give chlorine gas according to equations $(1)$ and $(2)$.
$$\begin{align}\ce{Cl- + OCl- + 2 H+ &-> Cl2 + H2O}\tag{1}\\
\ce{ClO3- + 5 Cl- + 6 H+ &-> 3 Cl2 + 3 H2O}\tag{2}\end{align}$$
This can happen if carbon dioxide is absorbed from the air by hydroxide according to $(3)$.
$$\ce{CO2 + OH- -> HCO3-}\tag{3}$$
However, you should be able to smell chlorine early enough if it starts to get dangerous. If upon opening the chlorine smell is stronger than usual, open windows immediately and leave the room.
All other decomposition products should not cause any great harm. However, chlorate’s oxidative ability is less than hypochlorite’s to the best of my knowledge, so you might find that the partially decomposed bleach does not clean as well as the fresh one.