Strongly real elements of odd order in sporadic finite simple groups

If my coding is correct, then the answer to your question is Yes: All real elements of odd order in the sporadic simple groups are strongly real, with the exception of 3a, 5a in McL. With GAP, it takes only about a second to check the tables.

The following GAP function returns the class position of all strongly real classes in a character table:

StronglyRealClasses:= function( tbl )
    local kG,         # nr conjugacy classes
          invs,       # class positions of involutions
          nrinvs,     # nr of involutions
          sreals,     # class positions of strongly real classes
          prodcls,    # class positions in a product of 
                      # two involution class sums
          i, j, k;

    kG:= NrConjugacyClasses( tbl );

    invs:= Positions( OrdersClassRepresentatives( tbl ), 2 );
    nrinvs:= Length( invs );

    sreals:= [ ];

    for i in [ 1 .. nrinvs ] do 
    for j in [ i .. nrinvs ] do

        prodcls:= Filtered( [ 1..kG ], 
                    k-> ClassMultiplicationCoefficient(tbl, invs[i], invs[j], k) <> 0 
                    );
        UniteSet( sreals, prodcls );

    od; 
    od;
    return sreals;
end;

(This is quite naive. You could modify this function by omitting the inner for loop and replace invs[j] by invs[i], to make the function more efficient for your intended application of odd order strongly real elements, as suggested in your comment. But since it takes only about a second on my desktop computer to apply the above function to all the sporadic tables, I did not bother to rewrite it.)

The following function returns the class positions of all real elements of odd order, but not strongly real:

RCoOOnSR:= function( tbl )
    local rc, ooc, src;

    rc:= RealClasses( tbl );
    ooc:= PositionsProperty( OrdersClassRepresentatives( tbl ), IsOddInt );
    src:= StronglyRealClasses( tbl );

    return Difference( IntersectionSet( rc, ooc ), src );
end;

(Type ClassNames( tbl ){ RCoOOnSR( tbl ) }; to get names of the classes.)

One more confirmation of OP's observation can be found in tables, from the paper:

Jordan Journal of Mathematics and Statisticscs (JJMS) l(2), 2008, pp. 97-103 97 STRONGLY REAL ELEMENTS IN SPORADIC GROUPS AND ALTERNATING GROUPS, IBRAHIM SULEIMAN

One can observe many real, but not strongly real classes, but most of them have even order, except group McL classes 3A, 5A - exactly as OP's proposal.

Table 3 will give the real classes which are not strongly real in Sporadic Groups

M11 all real elements are strongly real.
M12 all real elements are strongly real
M22 8A.
M23 8A
HS all real elements are strongly real.
J3 all real elements are strongly real.
M24 all real elements are strongly real.
McL 3A, 5A, 6A.
He all real elements are strongly real.
Ru all real elements are strongly real. 
Suz all real elements are strongly real.
ON all real elements are strongly real.
Co3 all real elements are strongly real.
Co2 16B.
Fi22 all real elements are strongly real.
HN 8A .
Ly all real elements are strongly real.
Th 8B.
Fi23 16AB, 22BC, 23AB.
Co1 all real elements are strongly real.
J4 all real elements are strongly real.
Fi_24 all real elements are strongly real.
BM all real elements are strongly real.
M 8C, 8F, 24F, 24G, 24H, 24J, 32A, 32B, 40A, 48A. 

Table 4 will give the complete list for all non real elements in the Sporadic Groups

M11 8AB, 11AB.
M12 11AB.
J1 None
M22 7AB.
J2 None
M23 7AB, 11AB, 14AB, 15AB, 23AB.
HS 14AB.
J3 19AB.
M24 7AB, 14AB, 15AB, 21AB, 23AB.
McL 7AB, 9AB, 11AB, 14AB, 15AB, 30AB.
He 7AB, 7DE, 14AB, 14CD, 21CD, 28AB.
Ru 16AB.
ON 31AB.
Co3 11AB, 22AB, 20AB, 23AB.
Co2 14AB, 15BC, 23BC, 30BC.
Fi22 11AB, 16AB, 18AB, 22AB.
HN 19AB, 35AB, 40AB.
Ly 11AB, 22AB, 33AB. 
Fi23 16AB, 22BC, 23AB.
Co1 23AB, 39AB.
J4 7AB, 14AB, 14CD, 21AB, 28AB, 35AB, 42AB.
Fi_24 18GH, 23AB.
BM 23AB, 30GH, 31AB, 32CD, 46AB, 47AB.
 M 23AB, 31AB, 39CD, 40CD, 44AB, 46AB, 46CD, 47AB, 56BC, 59AB,
62AB, 69AB, 71AB, 78BC, 87AB, 88AB, 92AB, 93AB, 94AB, 95AB,
104AB, 119AB.