How to extend path on only one side

Make two rectangles, group, then play with Object→Envelope Distort→Make-with-mesh (or even just Make-with-warp might be enough)...

Micha,

Now that different ways and means have been brought up, to elaborate on the option of starting over, if I were to make such a tricolore (the background being the third colour), I should create the entire outer shape first (more or less based on the existing red shape) to get the best overall appearance, then cut out the shape in the middle, then colour the remaining paths left and right.

Micha,

you told that you are rather new to Illustrator. That's just fine.

And at the same time it is one of the most important reasons to learn how to draw with the Pen tool. If you can passably handle this tool, you really do not need over-complicated ways or even plugins. You would just be able to draw it in 20 or 25 seconds.

Micha,

Always make backups of the artwork before any irreversible steps.

One crude way is to start over which implies a rough redoing of the red path; the original can be locked to use as a guide.

If you wish to keep the red path exactly as it is, you can smoothly continue the (leftmost) segments of the top and bottom sides of the red path, then create a path corresponding to the left side.

To continue the (leftmost) segment of the top side, you can (Smart Guides being your friends):

0) Make the blue rectangle much larger so you are free to trim it, you can also extend it to the right to overlap the red path, then Ctrl/Cmd+X+B to place it beneath the red path in the stacking order, then lock it;

1) Click the leftmost part of the top side of the red path, so the corner Anchor Point and the handle is revealed, then with the Line Segment Tool ClickDrag from the Handle into the Anchor Point (Smart Guides say handle and anchor when you are within snapping distance), then delest and ClickDrag the new line by its lower left end Anchor Point to snap to the corner Anchor Point;

Now you have the direction and length of a matching Handle which will ensure a smooth continuation;

2) If you wish to keep the curvature at the corner Anchor Point, skip to 3), if you wish to be able to extend it, click the lower left Reference Point in the Transform panel, then increase the W or H value (you can add *2 or something in the box) and Ctrl/Cmd+Enter to extend;

3) With the Pen Tool, Click and Hold close to the corner Anchor Point/end of line, then press the Spacebar and Drag to snap to the Anchor Point, then let go of the Spacebar and Drag along the line to the far end or as desired, then continue the path by at least one more Anchor Point to shape the top side of the blue path as desired, ensure that the new path extends past it; when drawn you can adjust with the Direct Selection Tool, and hide the line from 1);

4) Repeat 1) - 3) for the bottom side;

Now you have the shapes of the top and bottom sides of the blue path;

5) Trim the blue path by the new paths (you can use Object>Path>Divide Objects Below or Shape Builder or Pathfinder (Divide)), deleting the unwanted bits;

You can make the shape of the right side of the blue path as freely as in 6) or predetermined as you wish, ensure that the new path extends past the top and bottom of the blue path;

To have an equidistant (guide) following (of) the left side of the red path, you can:

5) Direct Select the segment that forms the left side (or hold Shift and select if more than one), then Ctrl/Cmd+C+X+F to bring it to the front (if more than one, Join them), then Object>Path>Offset Path using a, offset that corresponds to the desired gap, then Hide the original segment(s) and delete each straight end part of the offset path with the Direct Selection Tool (remember to deselect first), then delete the rightmost path;

6) If desired, you can edit the path from 5) or just use it as a guide for a new one;

7) Trim the blue path ad in 5).

Now you have all the curved shapes of sides of the blue path related to the red path; if needed/desired, you can create the shape of the left side as desired (straight or curved), and trim again.