How can I tell what was my tax paid to one state on double taxed income?

I started a job in a new state before I was able to move (I stayed in hotels while looking for a home) and moved to the new state a few weeks after starting the job. My W-2 has taxes paid to both states, the amount for my first state is a small portion of my overall wages, and the amount for the second state is the full amount of my overall wages for this employer. Does this mean I was double taxed on the portion of my income before I moved? If so, how can I tell how much tax was paid to the second state on this double-taxed income?

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