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. 

As the copyright holder of both the new code and the new combined work you may pick any license you want as long as in doing so you can also comply with the existing license. The recipients of the work from you must also be able to comply with both the existing license and your overlay license at the same time. But you are not “relicensing” the original work as only its owner can do that. Rather, you are adding an “overlay license”.

Some things to consider if you do this:

1. Do not remove any existing copyright statements. That could get you into a world of trouble. Add your own, sure, but don’t remove other people’s.
2. Consider the maintenance issues you may be causing. If your downstream wants to contribute to you upstream, will the upstream be able to accept those contributions from you as licensed? This is one of the biggest considerations; you may end up defacto maintaining a fork of upstream for lack of ability to contribute to it, which could be especially problematic when it comes to resolving upstream CVEs.
3. Make sure both licenses can be complied with at the same time. Just because you have overlaid a second license, that doesn’t mean you don’t have to comply with the requirements of the first any more. If they have mutually exclusive requirements that may be impossible. The obvious case is copyleft requirements, but it may also be impossible to comply with other terms, even perhaps the form of attribution. 
4. Your sense of style is not a reason to ignore requirements. If the first license says “this permission notice shall be included in all copies of substantial portions of the software” that remains a requirement even if the second license says something different or your project has a different “house style”. If you fail to do it, you no longer have a copyright license and are in breach of copyright. This all applies even if the two licenses are considered minimal. For example, to include MIT code in a BSD-3 downstream you still need to comply with the requirements of both and not remove existing copyright statements.
5. This all relates to cases where you are putting some code under a different license into a project you’re working on. If what you want to do is take a body of code under a minimal or non-reciprocal license and make it available essentially as is under a different license, you should seek legal advice from a law professional (if it’s not obvious, this practical-experiences article shouldn’t be considered legal advice).

So never say you are relicensing when you are not the copyright holder (or hold equivalent rights to relicense from, say, a CLA) as that term can lead you to assumptions and actions that could leave you in breach of copyright. You are most likely overlaying another license – why not say so?