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There has been growing interest in the statistics community to develop methods for inferring transmission pathways of infectious pathogens from molecular sequence data. For many datasets, the computational challenge lies in the huge dimension of the missing data. Here, we introduce an importance sampling scheme in which the transmission trees and phylogenies of pathogens are both sampled from reasonable importance distributions, alleviating the inference. Using this approach, arbitrary models of transmission could be considered, contrary to many earlier proposed methods. We illustrate the scheme by analysing transmissions of Streptococcus pneumoniae from household to household within a refugee camp, using data in which only a fraction of hosts is observed, but which is still rich enough to unravel the within-household transmission dynamics and pairs of households between whom transmission is plausible. We observe that while probability of direct transmission is low even for the most prominent cases of transmission, still those pairs of households are geographically much closer to each other than expected under random proximity.