I am having trouble finding a 3.4 µH through hole inductor that is in stock right now. I was wondering if I can just place 2 inductors that add up to 3.4 µH in series to solve this problem. I am not sure if this would introduce any problems, if so what kind of problems and how can I mitigate them.
Correct - putting inductors in series will result in an effective inductance equal to their sum.
The problem is the extra wiring, and/or change in the wires' paths. Moving two wires closer together makes them act like a small capacitor. Long wires act like a resistance, and (obviously) loops act like inductors.
3.4uH is relatively large, so you might not have much of a problem since inductors tend to have a large tolerance, +/- 10% is typical. Just keep the inductors close together, and modify the surroundings as little as possible.
There are other issues to consider though, such as internal resistance. (An ideal inductor has an internal resistance of zero ohms). Try to match the series resistance in the same way as inductance. Aim for the sum to equal that of the one you're replacing. This is going beyond my expertise, but I suspect it won't matter much; it depends on the circuit.
3.4 µH is not a standard value, but 3.3 µH is. There are plenty of those out there, even in thru-hole.
Inductors aren't generally accurate enough to justify making a 3.4 µH version when a 3.3 µH version is already in production. The extra 3% more isn't worth another product to manufacture, and for customers isn't worth another part to stock. You better have a really good reason you are committing your company to stock a new inductor with a non-standard value.
You need to step back and examine why you need such a unusual inductance value so accurately that 3% less actually matters. Usually you determine the rough inductance you need, find a suitable part, then design the details of the circuit around that. You can get resistors and timing components at much higher accuracies than inductors, for example, so you pick exact values for those last.
You should also examine why you want thru-hole. Unless this is a high-current high-power part that is therefore physically heavy, thru-hole makes little sense. Thru-hole will limit you to parts designed 10 years ago or longer. New development is obviously in surface mount parts, except for physically large inductors. You haven't said anything about the saturation current requirement or maximum resistance you can tolerate. "3.4 µH" is only one part of the spec for a inductor.