It appears you may be mistaken with the premise of your question.
A chain of command blocks will always continue and execute the full chain, unless said chain command blocks are set to the “Conditional” setting, highlighted below:
When any command block set to this setting is triggered, it will check the command block on the side directly opposite its arrow (not necessarily the command block that points at it), and it will only run itself if that command block's previous output was success.
But if that command block's previous output was a fail, then the conditional command block will not run its own command. However, if it is a chain conditional command block, it will still continue to propogate the chain signal down the line of chain command blocks, and any unconditional chain command blocks further down the line will still run.
Therefore, the most likely issue from the info you've provided is that your command blocks are set to “Conditional.” Setting the command blocks to unconditional will mean that they will always run when triggered, without caring about the previous output of any other command block.