I have been known to say that a company is a collection of processes. However, the combined processes are also part of a system. One process can affect another or others.
Applying an 80/20 rule, 20% of your processes will have the biggest impact on how your company functions. It is also very likely that many of these 20% processes will interact with others. That interaction could be unfavorable.
Realistically, all of your processes should work in concert with one another. So during your 'analysis' or 'discovery' phase, it is important to take into account how one process impacts another or others. Are people working within one process waiting for the output from another? What happens if that output is delayed?
Let's not forget your customer. It is important to understand how these processes interact with your customer. Your customers need to be not just happy, but ecstatic. A happy customer might still change suppliers, an ecstatic customer will not change and will provide referrals.
As an example - If we are talking about ecommerce, you need to make it very easy for your customer to make a purchase. You might want to look into doing extra things for your customer like Amazon did with 'Prime' and their one click purchase option.
I am not suggesting that you attempt to manage a bunch of processes within a single project, it is still best to manage/improve a single process at a time. I am suggesting that you take into account the adjacent processes that will need to be managed.
What has been your experience?