May 26 (Reuters) - ICE cotton futures rebounded from an over one-week low on Tuesday, as projections for lower rainfall in Texas, the largest cotton-producing state, eased concerns over increased supply.              
* Cotton contracts for July        rose 0.78 cent, or 1.35%, at 58.39 cents per lb. as of 01:13 p.m. EDT (1713 GMT). It traded within a range of 57.66 and 59.45 cents a lb.     
* "We were expecting more rain... It is dry in Texas and they are not getting the rain they need for a good crop," said Jordan Lea, senior trader at DECA Global.    * Also supporting the market, were hopes for a potential coronavirus vaccine and a pick-up business activity as economies reopen.                       * Demand for cotton and apparel has been hammered by worldwide coronavirus restrictions, sending prices of the natural fiber tumbling 15% so far this year.    
* The contract recorded its first weekly fall in five last week, as the U.S and China rift widened.      * "While there is some optimism around the new vaccine, we will also have to keep an eye on India exporting more cotton, taking advantage of the weak rupee," Lea added.    * A trade body raised its forecast for India's cotton exports on Monday as a fall in the value of the rupee         to a record low made shipments competitive.                 * Speculators trimmed their net short position in cotton futures by 1,125 contracts to 7,264 contracts in the week to May 19, U.S. Commodity Futures Trading Commission (CFTC) said on Friday.            
* Investors now eye a weekly cotton planting progress report due later in the day.    
* Total futures market volume fell by 1,346 to 17,318 lots. Data showed total open interest remained at 183,285 contracts in the previous session.    * Certificated cotton stocks CERT-COT-STX deliverable as of May 22 totaled 6,693 480-lb bales, up from 5,830 in the previous session.    (Reporting by Diptendu Lahiri and Nakul Iyer in Bengaluru; Editing by Bernadette Baum)