If you are looking for a secondary to cover Falco's bad match-ups, then you shouldn't really need one because he doesn't really have a significant disadvantage (if at all) against anybody.
If you are looking for a secondary to have more positive match-ups overall, then you have a few options. Falco generally struggles the most against Fox, Marth, Jigglypuff, Peach, himself, and maybe Samus, so find a character that does well or at least not poorly vs them. Characters that come to mind are Fox and Marth.
Fox does well vs every character and is a great choice in that sense; however, maintaining and warming up both is very difficult, so I generally don't think that it's worth it, on top of both characters having similar strengths in terms of stages and overall being redundant.
Marth works well because he does better vs Peach, Samus, and arguably Jigglypuff. Marth also does better on FD against Fox, Falco, and Marth than Falco does. Falco compliments Marth well because he deals with Sheik, Falcon, and probably Pikachu and Yoshi better than Marth does (not saying Marth loses to them, Falco is just better). Marth requires fewer button presses and offers a different perspective on the game for a Falco main such as strong ground movement, spacing, and choosing when to attack.
If you aren't worried about Fox, then Sheik also works decently because she is easy to maintain and to play and does better than Falco vs Samus, Peach, as well as being better on different stages than Falco. This does however leave something to be desired against Jiggs. Similarly, if you don't care about Fox, Jigglypuff can work better vs Samus and Peach. This also eliminates any potential counterpicks from an opposing Jiggs player on top of offering a niche vs unsuspecting Falcos.
The rest of the cast has very little to offer as opposed to the top tiers as far as match-ups go. I don't really think that Falco should need a secondary and is completely viable to solo-main, but if you really want one, then I recommend Marth. Sheik and Jiggs each offer a niche, and Fox is great, but it's just too hard and not worth it to play both.