Can I start a new uart1 interrupt function inside a previous uart1 interrupt complete callback function?
Sure, just make sure to avoid race conditions and blocking in anything called from an ISR.  For example, if HAL_UART_Transmit_IT blocks for buffer space to become available, you must not call it from an ISR as this will almost certainly cause a hang.  Also, HAL_UART_Transmit_IT must be reentrant as your interrupt could fire while HAL_UART_Transmit_IT is executing in your main loop.  Another option would be to set a flag and handle the transmission somewhere else.