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I have finally found the following papers and results, which predate Nagura's paper of 1952. I cite them from newest to oldest:

1. (Molsen, 1941):

• For $n\geq 118$ there are primes in $(n,\frac43n)$ congruent to 1,5,7,11 modulo 12.
• For $n\geq 199$ there are primes in $(n,\frac87n)$ congruent to 1,2 modulo 3. This result implies that of Nagura.

K. Molsen, Zur Verallgemeinerung des Bertrandschen Postulates, Deutsche Math. 6 (1941), 248-256. MR0017770

2. (Breusch, 1932):

• For $n\geq 7$ there are primes in $(n,2n)$ congruent to 1,2 modulo 3 and to 1,3 modulo 4.
• For $n\geq 48$ there is a prime in $(n,\frac98n)$. This result implies those of Nagura and Molsen (but not the congruences part).

R. Breusch, Zur Verallgemeinerung des Bertrandschen Postulates, dass zwischen x und 2x stets Primzahlen liegen, Math. Z. 34 (1932), 505–526. MR1545270

3. (Schur, 1929, according to Breusch in the previous paper):

• For $n\geq 24$ there is a prime in $(n,\frac54n)$. This result already implies those of Hanson and El Bachraoui.

I. Schur, Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen I, Sitzungsberichte der preuss. Akad. d. Wissensch., phys.-math. Klasse 1929, S.128.