The example that provoked the question was hearing Marco Wyss talk about his book Postcolonial Security: Britain, France, and West Africa's Cold War. Wyss mentions that he faced a challenge because British and French state archives retain an almost unmanageable overabundance of material on colonial and post/neo-colonial policy in Nigeria and Côte d'Ivoire respectively, but archives in those African countries themselves suffer from a challenging lack of high quality sources.
It seems straightforward to think of other cases where a power imbalance might lead to a similar situation: the genocide of a demographic minority population by the machine of the nation state, conflicts of conquest between large powers and very small ones, etc.
Wyss says that he adopted a "traditional" method, of relying heavily on British and French sources, but reading them through a critical lens to try and bring out African voice and agency. This would suggest there are other, non-traditional approaches. I would love to hear how different historians handle these challenges in their research.
This is a little far afield from your West African example, but you might enjoy this post I wrote on resistance and accommodation in the mission system along the northern frontier of the Spanish Empire. In my area of expertise, an interdisciplinary approach that combines archaeology, bioarchaeology, and indigenous history complements the historical record. The missions are popularly viewed as one of the most oppressive environments imaginable. Archaeology and indigenous history show the processes of adaptation, accommodation, and resistance nearly invisible in the written documents. This interdisciplinary approach allows us to tell a fuller story of life in the missions, restoring voice and agency to those who lived in a highly oppressive system.
The traditional method, as Wyss terms it, is archival research. You look through state and para-state documents to piece together history. Saying he is trying to use a critical method honestly does not mean much to me, as it seems the vast majority of historians (excluding some hyper-empiricist historians often focused on political or military history) never accept a document for granted but instead try to understand the unwritten insights as much as the written insights.
A non-traditional method could be one one of two ways: either utilizing different sources, or taking a more theoretically-rigorous approach to the sources. I assume by method Wyss is referring to the sources consulted, so I will stick to that. Beyond the ones mentioned already by others: Anthropological sources are an incredibly important, and many historians -- especially those versed in Area Studies -- readily utilize them. Geography can also be a source, as can scientific data (especially for environmental historians). Art is a source. Even outside official archives exists a staggering number of documents which never end up in official archives (of if they do, tend to be overlooked), like receipts. Jamie Monson in Africa's Freedom Railway merged traditional archival sources with her own anthropological fieldwork and an abundance of receipts and bills-of-lading from companies, shippers, and receivers -- stuff not normally retained by archives.
In short, anything can be a source; it depends how you choose to analyze it and what you want the source to do.
You might look into subaltern studies, a sub-field of postcolonial studies. This is a good overview: https://scholarblogs.emory.edu/postcolonialstudies/2020/02/17/subaltern-studies/