Depends on the language. This answer provides an example in HLASM and a reference to the 'C' Language reference at the end.
For Assembler you'll need to REWIND when you CLOSE the DCB. See the label at FINISH to see how this is done.
There may be other ways but this worked for me on z/OS 2.4
PRINT NOGEN
* ------------------------------------------------------------------- *
* *
* SYSIN2 *
* *
* @author Hogstrom *
* *
* Test to see if one can re-read SYSIN more than one time. *
* *
* ------------------------------------------------------------------- *
R0 EQU 0
R1 EQU 1
R2 EQU 2
R3 EQU 3 * Number of times to loop
R4 EQU 4
R5 EQU 5
R6 EQU 6
R7 EQU 7
R8 EQU 8
R9 EQU 9
R10 EQU 10
R11 EQU 11
R12 EQU 12 * Base Register
R13 EQU 13
R14 EQU 14
R15 EQU 15
*
SYSIN2 CSECT
STM R14,R12,12(R13)
LR R12,R15
USING SYSIN2,12
*
ST R13,SaveArea+4
LA R0,SaveArea
ST R0,8(R13)
LA R13,SaveArea
*
OPEN (SYSIN,(INPUT))
OPEN (SYSOUT,(OUTPUT))
LA R3,3 Number of times to read SYSIN
*
GETAGAIN DS 0H
GET SYSIN,INREC Read the next rec
PUT SYSOUT,INREC Write that bad boy out
B GETAGAIN
FINISH DS 0H Invoked at End of Data of SYSIN
CLOSE (SYSIN,REWIND) Close SYSIN and rewind
OPEN (SYSIN,(INPUT)) Open it up again
BCT R3,GETAGAIN Repeat the madness
*
CLOSE SYSIN
CLOSE SYSOUT
*
L R13,SaveArea+4
LM R14,R12,12(R13)
XR R15,R15
BR R14
*
SYSIN DCB DSORG=PS,MACRF=(GM),DDNAME=SYSIN,EODAD=FINISH, *
RECFM=FB,LRECL=80,BLKSIZE=0
SYSOUT DCB DSORG=PS,MACRF=(PM),DDNAME=SYSOUT, *
RECFM=FBA,LRECL=133,BLKSIZE=0
*
INREC DC CL132' '
SaveArea DS 18F
LTORG
END
Assemble the above and execute this JCL
//SYSIN2 JOB (CCCCCCCC),'HOGSTROM',
// MSGLEVEL=(1,1),
// MSGCLASS=O,
// CLASS=A,
// NOTIFY=&SYSUID
//*
//STEP1 EXEC PGM=SYSIN2
//STEPLIB DD DSN=USER1.TEST.LOADLIB,DISP=SHR
//*
//SYSIN DD *
REC 1 OF 3
REC 2 OF 3
REC 3 OF 3
/*
//SYSOUT DD SYSOUT=*
//SYSUDUMP DD SYSOUT=*
And you'll get this output

You'll see the same SYSIN data repeated three times.
For a 'C' program this reference in the IBM documentation for the C Compiler provides some hints about rewinding.
To open an in-stream data set, call the fopen() or freopen() library
function and specify the ddname of the data set. You can open an
in-stream data set only for reading. Specifying any of the update,
write, or append modes fails. Once you have opened an in-stream data
set, you cannot acquire or change the file position except by
rewinding. This means that calls to the fseek(), ftell(), fgetpos(),
and fsetpos() for in-stream data sets fail. Calling rewind() causes
z/OS XL C/C++ to reopen the file, leaving the file position at the
beginning.
What you're looking for is possible but the implementation is language dependent.