Fred, further details are necessary for us to understand the situations to offer advice or help. For instance what was the initial alpha value before the reversing, item dropping and recoding? What was the value after this action? We can be on the same page and offer suggestions when this is given as well.
Yea. as fred said, it is important to for us to understand the initial value in order to really appreciate the situation and advise appropriately.
That notwithstanding, I've ever experienced a similar situation. I my case, the value even dropped significantly after taking out some of the items and reverse coding others.
My final resort, however, was that I collected more data and when I run again, it looked good. So, you may consider increasing your sample size and see. It could make it look better. Hope it helps.
As we wait for further particulars, I must add that the number of scale items also influence the alpha value you will obtain. Fewer scale items may result in low Cronbach alpha values and vice versa.... Consider that as well