You could simply use the word awkward. It's broader than cases where "a previously used word makes the latter obvious," but it still applies.
Awkward (abbreviated as awk) is part of the standard editor's/proofreader's notation. One website calls it a catchall term. It gets defined in various ways, such as:
- I can understand this, but it's not well said 1
- Awkwardly expressed or constructed 2
- The phrase is grammatically correct, but there is a clearer, more concise way to phrase it 3
- This sentence or construction is awkward; it should be written differently 4
I think all of those would apply to your "all the employees except for some exceptions" example.
Interestingly enough, one website provides an example of an "awkward expression or construction," and it's not too far off from the example in your question:
The storm had the effect of causing millions of dollars in damage.
Much like in your example, "had the effect of causing" is somewhat redundant, although the wording may not be quite as glaring as "except for some exceptions." In any case, the fix seems pretty straightforward – a simple shortening of the sentence will do the trick:
The storm caused millions of dollars in damage.