Since you left your 6-mth rehab job for the med-surg hospital job that didn't work out, this is what I would say in interviews when questioned about your 2 short-term positions. I'd say that I was interested in both rehab and med/surg, but thought at the time that the rehab position was the best fit. After a few months, I felt I wasn't being challenged enough and realized that I was more interested in med/surg and was fortunate to obtain a hospital position. But that the med/surg unit where I worked was not a good fit, due to some factor like lack of support or something else that is not too revealing and doesn't reflect poorly on either yourself or the previous employer. Whatever reasons you come up with need to fit the job you are applying for. This wouldn't work if you were interviewing for another rehab job or tried to return to your previous rehab job, unless you decided to say that med/surg was too fast-paced, etc. for you and so you realized rehab was your current niche.
I know fairly new nurses who obtained hospital med/surg jobs and psych facility jobs after having only LTC experience, but they had at least a year's experience in one position.
If you are applying on-line to positions, you usually can update/change your resume. If you are applying to a different unit with a different manager, your previous applications/interviews at that hospital probably won't hurt you. I imagine since you were not hired, they would throw out any paper resumes/applications/notes, at least after 3 months. If you take a LTC job now and later apply to a hospital, you can explain that you had difficulty getting a hospital job previously due to your lack of experience. Whatever you do, try to stay in the next job for at least a year.
To all new grads reading this - I think this nurse's experience could be a wakeup call to try if at all possible to stay in your first position for at least a year, even if it is not your first choice and you get an offer for what appears to be a better position. You never know how a new job will turn out, and if it doesn't work out and you get a history of short-term jobs without any long-term nursing jobs, you may very much regret having jumped ship due to great difficulties getting interviews/offers. Job opportunites expand greatly once you have a solid 1-2 years of experience in one job.