Looking at that code, it doesn’t look like there are any filters that would allow you to do what you’re hoping. This isn’t particularly surprising…
Taking a step back, Posts are inherently intended to contain chronologically-ordered content, so it’s unusual for plugins to allow ordering of posts by anything other than their published date. If you’re needing to do this, I’d strongly consider finding a solution where your content is stored in pages or a custom post type. Pages, for instance, have a field called Menu Order that is exactly intended for what you describe. This field is not available on Posts. (Note: It’s possible that Post Types Order would let you do this, but then the order of posts you set would be reflected everywhere, not just in the widget.)
I’d recommend finding a different widget that allows this. The one I know off the top of my head is Query Posts, though it would be cumbersome. You’d collect and order the IDs of each post you want, and enter them in the post__in field of the widget settings. Then you’d set the orderby field to post__in. (Flexible Posts Widgets may also allow this.)