If you'd like to have a card available to you with a lower APR, then the easiest time to get that is when your credit profile doesn't suggest to creditors that you need it.
If you ask for a reduction, make sure they don't mistake this as a request for help with a hardship. It needs to just be a permanent reduction in the ongoing APR. If you ask for the reduction on an account that doesn't have a balance then I don't think there's any risk of that being misunderstood.
Besides asking for a reduction, another way to get a lower APR is to just apply for another card. If your credit is currently in a good state for applying, then this might give you a significantly lower APR than what you'll get asking for a reduction on one of the existing accounts. It seems like lenders are far more resistant to lowering APRs than they are to raising limits, for whatever reason. Lots of creditor web sites make it easy to ask for CLIs, but I've never heard of any that have a "lower my APR" button.
Prequal sites might give you an idea of what APR you can expect. If you apply directly in response to a prequal, then I think you might be guaranteed that rate (or else they decline it). Somebody can correct me if that's wrong.
Prequals might even tell you what will happen with a cold app. I recently checked for Chase prequals and they offered a FU at the low end of the advertised APR range. I didn't want the FU though, I wanted the 5% Freedom. I worried that a cold app would never receive the lowest rate, but I did it anyway, and to my surprise they did give me the same rate that the FU prequal had offered.
I don't think Chase is the best for low APRs, but I'm just saying that prequals can be informative of what to expect from a lender. For some reason Citibank didn't like me much, their prequals had very high APRs. CapOne was the lowest for me. You can use the prequals to shop around.
But if this isn't a good time for applying, then don't. You'll get an optimal APR by applying at an optimal time.
