Can I refinance after 6 months?

Can I refinance my home after 6 months? I have about 40,000 in equity(large down payment). Then I lost my job. I need to payoff debt and buy a cheap car. Credit is medicore. I was told I could not get a home equity line of credit because of the amount of time we have lived in the house.
Update: I forgot to mention I have a job now. I recently had it appraised. 40,000 is a low estimate of the equity we have in the house.
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