The issue (I think) you face is a lack of knowledge about the size and location of the refugee population. This means that you may need to use a novel sampling strategy to survey this population. A couple of approaches spring to mind. These are Time-Location-Sampling (TLS) and Respondent-Driven-Sampling (RDS).
TLS : It is often possible to identify locations and times at which you can find refugees. You can sample these directly or you can use semi-quantitive methods (e.g. interviews, group discussions) to inform further sampling (i.e. identify clusters to sample). You can also use the TLS approach to locate RDS "seeds" (see below).
RDS : This approach relies on (first) identification of well-connected "seed informants" who identify small sets of survey respondents. Each of these respondents then identify a small number of further survey respondents. Data collection follows chains of identified respondents. With this approach it is essential to pick a good set of seeds which capture variability in the population of interest. Keep the set of survey respondents identified small (e.g. a respondent identifies < 5 further respondents), and allow the survey to run for a reasonably large number of rounds (something like 8 to 10 links in the chain - this can be helped by limiting the size of sets).
Both methods are used to sample from "difficult to sample" populations such as MSM, CSW, IVDU in a wide variety of settings.
These methods are not without problems regarding coverage (this can be fixed by good selection of the initial seed respondents and allowing sufficiently long chains to explore the population of interest in RDS and by a fairly exhaustive set of times and locations for TLS) and loss of sampling independence. The TLS sample (either direct or indirect) cane be treated as a cluster sample with posterior weighting based on some estimate of cluster populations (only relative sizes are required). The RDS sample consists of small clusters of social networks embedded inside a set of larger social networks. Classification techniques such as LQAS are quite robust and can be used. Estimation is a little more difficult but statistical / mathematical techniques have been developed and software is available. You need to make sure that you dataset include "metadata" that can identify who referred who so that the chains of referral can be identified (some reading will be required to get this right).
WRT the host population ... you could use a SMART-like (PPS) sample or RAM (spatial first stage sample) which are broadly equivalent methods in terms of results (i.e. estimates and precision). RAM costs about 50%-60% the cost of SMART and needs about 50% of the time.
All this assumes that you want population-level estimates (or classifications) rather than mapping. If you need advise on samples for mapping then let me know through this forum.
I can put together a small literature pack on RDS and make it available for download if needed. Again, let me know if this would be useful through this forum.
BTW : If you have a well-mixed and host populations than a single survey with a spatial first stage sample might suffice.
I hope this is of some use.