I also experienced a huge difficulty spike in the game when the above mentioned aliens first appeared. It was at the time when I felt in control, I had great equipment and I was steamrolling the aliens in every battle. I got utterly destroyed in my first battle when the aliens returned with better forces.
However, I think I've found a way to counter it. The trick lies in how you use reaction fire.
. UFO
.
.
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@##
#: wall
@: your soldiers facing North
UFO: expected enemy position.
... : path the aliens take
With the above RF strategy, you will get on at the beginning with some casualties, and by the mid-game with some new researched weapons you might even feel invincible, defeating the aliens every time with no or few casualties. And when the aliens begin to send better troops, you will fail miserably.
Why? Because the aliens spot you the same time as you spot them! They try to fire their guns, receive exactly one RF from each of your solders which at this time is not enough to kill them, then unload their full magazine of needles into you, wiping out most of your squad in one turn.
... UFO
...
...#########
... #########
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@##
In the above case, the aliens walk trough your field of vision, and receive reaction shots just by walking! Even if they spot you, they are already wounded and further reaction shots kill them with high probability.
If you don't have buildings with the features depicted above, you can create this situation with smoke grenades. I never used them before, but they proved to be very valuable for this job. Even if your situation is not exactly as I depicted, you can see the general idea: let the enemy walk through your field of vision, rather than into your field of vision from the front.