Is it possible to change the way the Blend tool works here?
What if I need the Blend tool draw the crossing not under the cross but above? Right where I drew the red lines on the screenshot.
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The Blend tool essentially looks at the area between the objects to be blended and fills that area with the blend. What you are asking is if the blend tool could create elements outside the area encompassed by the blended objects. That is not possible.
Basically if you were to draw a shape which touches the corners of every area to be blended, that's the area the blend tool fills.
That being posted, you may find different results based on front to back objects. Reversing stacking order can sometimes alter the blend. In addition, you can draw a path you'd like a blend to follow then use the Replace Spine option to change the default blend path.
But I don't think you can get what you've indicated via a blend unless you alter one of the blended paths. Moving the blue line further left would fill that area you've indicated a bit more.