Does the expression above means "(increased) more than three times" or "around three times"?
No. 翻三倍,means to octuple, 2^3, which is 8, not three times here; when you add 多, it means over the 8 times. Be very careful when you use 翻 in Chinese. It doesn't mean multiply, but power. For example, 翻一倍, means double; 翻两倍 means quadruple, 4 times.
Do I have to say "翻了三多倍" to mean the latter?
No. It is not right to say 翻了三多倍。多 is placed in the end here; You can use 多 when the number right before 多 is at least 10. For example, 我去了十多次医院。I have been to the hospital for over 10 times. 那是一个十多岁的小女孩。She is a little girl aged over 10. However, for time, we don't have 十多点 for time between 10 o'clock and 20 o'clock. We say 十点多 to refer time over 10 and before 11. 3 is less than 10, so you use 多 after 三倍。