That is a good question and I do not know the answer. This article (Two-promoter vector is highly efficient for overproduction of protein complexes
Kyung-Jin Kim,1 Han-Eol Kim, Kwang-Hoon Lee, Wondeok Han, Min-Ju Yi, Jinseok Jeong, and Byung-Ha Oh) discussed whether we should use TWO promoters for the expressions of TWO genes. While whether two promoters in front of a single gene would enhance or reduce the expression of one single gene seems hard to answer for me.
If so, I would put the more ubiquitous promoter second as there might be carry over from the first. If you have a polyA tail after the first one that might disrupt transcription of the second. But in a lot of cases like a lentiviral vector you don't want pA tails in the first place.
Yes, you might get higher expression of the cloned fragment than you may want as the first promoter may still have the read through activity. I had the same issue few years back with my cloning. Luckily I found a restriction site between the promoters and cloned my interest of gene between the promoters to avoid the over expression. Other hectic option is if the vector has one restriction site before the first promoter and most likely the second restriction site next to the second promoter, then may remove both the promoters and clone the gene with it's own promoter or with the vector promoter, but its very time consuming and challenging. All the best.