EK-TM4C129EXL: Forward SPI frame

Yes, use advanced mode, with hold frame enabled. You send all bytes with SSIDataPut() except the last byte you send with SSIAdvDataPutFrameEnd(). To get the data to pass from one slave to the next, it will be necessary to read the received data and, if FSS is not deasserted, put it into the output buffer so it will be sent to the next slave. This will need to be done before the master starts transmitting the next byte. I have to admit that I do not have time to write this code and verify it, but here is what I think it looks like to initialize a master (example uses SSI1) and a slave (example uses SSI3).

#define MASTER SSI1_BASE
#define SLAVE SSI3_BASE

// Initialize SSI1 as master
void QSSIMasterInit(uint32_t SysClk_Freq, uint32_t baud_rate)
{
	MAP_SysCtlPeripheralEnable(SYSCTL_PERIPH_SSI1);
    MAP_SysCtlPeripheralEnable(SYSCTL_PERIPH_GPIOB);
    MAP_SysCtlPeripheralEnable(SYSCTL_PERIPH_GPIOE);

    MAP_GPIOPinConfigure(GPIO_PB5_SSI1CLK);
    MAP_GPIOPinConfigure(GPIO_PB4_SSI1FSS);
    MAP_GPIOPinConfigure(GPIO_PE4_SSI1XDAT0);

    MAP_GPIOPinTypeSSI(GPIO_PORTB_AHB_BASE, GPIO_PIN_4 | GPIO_PIN_5);
    MAP_GPIOPinTypeSSI(GPIO_PORTE_AHB_BASE, GPIO_PIN_4 );

    MAP_SSIConfigSetExpClk(MASTER, SysClk_Freq, SSI_FRF_MOTO_MODE_0, SSI_MODE_MASTER, baud_rate, 8);
    MAP_SSIEnable(MASTER);
    SSIAdvModeSet(MASTER, SSI_ADV_MODE_READ_WRITE);
    SSIAdvFrameHoldEnable(MASTER);
}

// Initialize SSI3 as slave
void QSSISlaveInit(uint32_t SysClk_Freq, uint32_t baud_rate)
{
    MAP_SysCtlPeripheralEnable(SYSCTL_PERIPH_SSI3);
    MAP_SysCtlPeripheralEnable(SYSCTL_PERIPH_GPIOQ);

    MAP_GPIOPinConfigure(GPIO_PQ0_SSI3CLK);
    MAP_GPIOPinConfigure(GPIO_PQ1_SSI3FSS);
    MAP_GPIOPinConfigure(GPIO_PQ2_SSI3XDAT0);

    MAP_GPIOPinTypeSSI(GPIO_PORTQ_BASE, GPIO_PIN_0 | GPIO_PIN_1 |GPIO_PIN_2 );

    MAP_SSIConfigSetExpClk(SLAVE, SysClk_Freq, SSI_FRF_MOTO_MODE_0, SSI_MODE_SLAVE, baud_rate, 8);
    MAP_SSIEnable(SLAVE);
    SSIAdvModeSet(SLAVE, SSI_ADV_MODE_READ_WRITE);
    SSIAdvFrameHoldEnable(SLAVE);
}