The custom cycle selector can break up a single trace into multiple traces
1. Use a different color to highlight when things go off the rails
Use a custom cycle selector to break the above trace into two traces. The first trace is cycles 1 through 399, and the second trace is cycles 401 through 1000. Then, apply a different formatting, like the color red, to the second trace.
This makes removes ambiguity on how to interpret the data and makes it visually distinct where the separation is.
2. Change the marker, size, or opacity of outliers so they don't detract from the big picture
Using the same strategy as above, a custom cycle selector can break the above trace into multiple traces: cycles 1-50, cycles 52-53, and cycles 55-100. Since cycles 52-53 contain the outliers, formatting can be applied that makes it smaller or transparent, so that it doesn't detract from the big picture.
3. Completely remove the outliers, and use an annotation to explain
If it's not desirable to show the outliers at all, they can be removed by either setting the opacity to 0% in the formatting, or using a cycle selector to exclude the offending cycles. It can sometimes be helpful to create a text annotation in the chart to explain that the outliers were removed.