You want to manage your havens in order to get intel, through the command menu of the avenger. This will allow you to be proposed missions with sufficient infiltration time. Infiltration time is high when you are informed as early as possible about the possibility of a mission. Of course, as long as you are proposed missions with too many ennemies, you just refuse them and you wait for the right opportunity.
You need to choose the missions where you send your soldiers and that means advance notice, that you get through managing the havens, mentioned hereabove.
This is reasonably well explained on the LWOTC Wiki as well as in the help available in the game itself.

I've got 13 people doing intel and still get a day or a day and a half.

Even with 13 people on intel you are not garuanteed to detect the supply raid missions in time. Your chances are good, but it doesn't always cut it.

Supply raid missions are among the hardest missions to detect with naturally low time to infiltrate them (the timer starts at 5 days I believe).

What you can still do to improve your chances of detecting them is setting a scientist as the haven advisor or installing a radio relay. You can also do the advent comms tower liberation mission to get a radio relay for free. To get that mission you will need to do 3 other liberation missions first though (in order: "start liberating the region" -> "rescue a defector" -> "hinder advent operations", these descriptions can be found in the mission screen when you select it).

I didn't realise you get a free radio tower when you do a comms mission. That's good news. Thanks for the advice.