Using open source elsewhere is not relicensing, it’s overlaying a second license.

So you’re considering taking some open source code under a minimal, non-reciprocal OSI-approved license and putting it under a different open source license, hopefully in combination with your original code (or another form of larger project). 

Don’t call this “relicensing” – it is not! The original license will continue to apply and you remain responsible for complying with its requirements. Only the copyright holder can change the license. You’re not relicensing – instead you are using the rights the license has given you and applying an additional license to the combination of the earlier work and your work.