Most do not, BUT I have seen Southwest do a yearly sale and Spirit Airlines has their 'club' you can join to get reduced airfare year round. The club does cost a fee but the thing I like about the Spirit site is that it will show you what the 'club' price would be so you can compare if it's worth it for your trip.
If you are on Twitter, you can follow the airlines. If you don't want them to appear in your feed every time you check Twitter, you can create a list and put the airlines in there. That way you can check the list when you want.
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