Recent theoretical results have shown that instilling knowledge transfer into black-box optimization with Gaussian process surrogates, aka transfer Bayesian optimization, tightens cumulative regret bounds compared to the no-transfer case. Faster convergence under strict function evaluation budgets - often in the order of a hundred or fewer function evaluations - is thus expected, overcoming the cold start problem of conventional Bayesian optimization algorithms. In this short paper, we prove that the regret bounds can be further tightened when extending the method to multi-source settings (where each source may depict distinct source-target correlation), while also achieving internal algorithmic complexity that's only linear in the number of sources. Experimental results, on both synthetic test functions as well as a practical case study, verify our theoretical claim of performance gain.