BENGALURU, Dec 29 (Reuters) - Indian shares hit another record high on Tuesday, extending their rally backed by foreign inflows, with private-sector lenders gaining the most due to an improving economic outlook. The NSE Nifty 50 index was up 0.57% at 13,953.10 by 0354 GMT, while the S&P BSE Sensex was 0.6% higher at 47,644.29. Both indexes hit record highs. Lenders HDFC Bank Ltd and ICICI Bank Ltd were the top boosts to the Nifty. In encouraging news on the COVID-19 vaccine front, Indian vaccine maker Serum Institute said it expected the British and Indian governments to approve the Oxford/AstraZeneca shot for emergency use within a few days. Hopes that a U.S. pandemic relief package would be expanded were aiding investor sentiment globally, boosting broader Asian shares. (Reporting by Chandini Monnappa in Bengaluru; Editing by Vinay Dwivedi)