Evelynn scales pretty hard into the lategame, actually. She's got some really high AP ratios. If your issue is closing off long matches, I'd suggest running Gathering Storm because you just keep scaling. One thing to note, is that you're very team reliant when playing as Evelynn. Having a good frontline is a must.
Evelynn is best on a team with a high damage mage and a strong frontline, since your MR shred is highly valuable even if you fall behind and aren't able to safely pick people off. If you don't have a good frontline to make the fights happen, you're going to have a rough time, as having a good frontline will enable you to assassinate one, two, or sometimes even three squishies in a good fight.
Not having a strong frontline means you're likely going to have a very bad time as Evelynn, since you'll be required to make some seriously high risk plays, and you won't have much disruption available to keep you safe when you're going for the kill.
In general, your goals as Evelynn in the lategame are:
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Pick off enemy Marksmen and Mages. Mages can be higher priority targets than Marksmen, Brand for example, so don't limit yourself to focusing down Marksmen.
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Pick off enemy Assassins and Squishy Supports. Taking out Soraka, Sona, Janna, or Lulu can sometimes be more impactful than taking out the Marksmen themselves. Opportunistically try to take out enemy Assassins, as it will protect the rest of your team's squishies, if they're not significantly ahead of you.
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MR Shred Fighters and Juggernauts for magic damage teammates if you're unable to pick off any squishies.