Confronting a new host involves new selective preassures that must be overcome by the viral population, meaning that the virus will be forced to change rapidly to get adapted to the new environment. These selective constraints cause that, when a host-jump takes place, the genetic diversity of the viral population is affected mostly because of population bottlenecks (not every individual in the viral population will be able to efficiently propagate in the new host, and only the mutant spectrum that includes the required mutations will be established). The new host presents to the viral population a new fitness landscape, and the viral population might should find new ways to explore the sequence space. That would cause that, when the population size is recovered in the new host, the mutant distribution of the viral population is different to that one infecting the previous host. The ability of an organism to effectively explore the sequence space is the evolvability, that in viruses depends mainly on the replication fidelity of the replicase (which in your case would be interesting to analyze in order to search for substitutions affecting the mutation frequency of the population). Anyway, changes in evolvability are not required, since you can observe an increase in the evolution rate only due to an increase of the rate of selection.
My answer to your question would be yes, it is possible (and convenient too, I would think). Nevertheless, there are many factors that must be taken into account, like the type of virus you are working with and the function of the gene you are studying. Also, I don´t think that the rate of evolution would remain high in the new host for a long time, just through the adaptation process.
I hope this help rather than confuse.
For further readings I would suggest:
Domingo, E. (2010) Mechanisms of viral emergence. (Vet Res.)
The answer depends a lot on how much difference there is between the two host species. For example, HIV-1 came to humans from Chimpanzees while HIV-2 came to humans from sooty mangabeys. The SIV-SMM to HIV-2 "jump" was far larger than the SIV-CPZ to HIV-1 "jump". Both of these viruses have crossed the species barrier several times, the SIV-CPZs created HIV-1 M, N, O and P groups, while the SIV-SMMs created HIV-2 groups A, B, C, D. In both cases, only one of the several cross-species jumps caused serious epidemics in humans, the HIV-1 M group and the HIV-1 A group. The HIV-2 A group has not caused nearly as big of a human epidemic as the HIV-1 M group; the HIV-2 A epidemic is more similar in number of humans infected, to the HIV-1 O group epidemic.
Most viruses cannot make a truly huge species jump, such as from a primate to an herbivore, or from a feline to a primate. There are lentiviruses (HIV-1 and HIV-2 are lentiviruses) that infect sheep, goats, cats, horses, cattle, etc... The sheep lentivirus can infect goats and the goat lentivuruses can infect sheep. But thee is no evidence for larger cross-species jumps. Most of the SIVs from other primates are not able to productively infect human cells, and much of the biochemistry of what the cross-species restrictions are have been worked out by swapping host and viral genes.
The rates of evolution vary more between modes of epidemic spread in humans, than between human and chimpanzee or between human and sooty mangabey. When the virus is rapidly passed from one human to another via dirty needles in IV drug abuse outbreaks the evolution of the envelope gene is slower because the virus is not driven to evolve by each host's immune response before it is passed on to the next individual.
See for example:
The evolutionary rate dynamically tracks changes in HIV-1 epidemics: application of a simple method for optimizing the evolutionary rate in phylogenetic trees with longitudinal data.
Maljkovic Berry I, Athreya G, Kothari M, Daniels M, Bruno WJ, Korber B, Kuiken C, Ribeiro RM, Leitner T.
Epidemics. 2009 Dec;1(4):230-9. doi: 10.1016/j.epidem.2009.10.003. Epub 2009 Nov 12.
PMID: 21352769
Unequal evolutionary rates in the human immunodeficiency virus type 1 (HIV-1) pandemic: the evolutionary rate of HIV-1 slows down when the epidemic rate increases.Maljkovic Berry I, Ribeiro R, Kothari M, Athreya G, Daniels M, Lee HY, Bruno W, Leitner T.
in the case of flu, the recombination between two lines of viruses (presumably avian-type and mammal-type) simultanously present in the same host (pig) could have been the origin of the extremely virulent 1918 pandemic ; this is a case of rapid evolution possible since the host accepted viruses from different origins, a property that may be is not shared by all different hosts affected by the different viruses of this family of viruses