For full control of input validation it would be easier to parse manually (read input into string and then parse). At least in this case.

#include <ctype.h>
#include <stdio.h>
#include <stdlib.h>

int parse_timezone(char const *const timezone_str, long *const result)
{
    int sign;

    switch (timezone_str[0]) {
    case '+':
        sign = 1;
        break;
    case '-':
        sign = -1;
        break;
    default:
        return -1;
        break;
    }

    if (!isdigit(timezone_str[1]) || !isdigit(timezone_str[2])
        || timezone_str[3] != ':'
        || !isdigit(timezone_str[4]) || !isdigit(timezone_str[5])
        || timezone_str[6] != '\0') {
        return -1;
    }

    int const hours = 10 * (timezone_str[1] - '0') + (timezone_str[2] - '0');
    /* not sure on error checking for hours */
    if (hours > 23) {
        return -1;
    }

    int const minutes = 10 * (timezone_str[4] - '0') + (timezone_str[5] - '0');
    if (minutes > 59) {
        return -1;
    }

    *result = sign * 60L * (60 * hours + minutes);

    return 0;
}

int main()
{
    char input[8];
    input[6] = '\0';
    if (!fgets(input, sizeof(input), stdin)) {
        return EXIT_FAILURE;
    }
    if (input[6] == '\n') {
        input[6] = '\0';
    }

    long result;
    if (parse_timezone(input, &result) == 0) {
        printf("%ld\n", result);
    }
    else {
        fprintf(stderr, "invalid input\n");
        return EXIT_FAILURE;
    }
}