The field strength of a solenoid is proportional to (N×I)
2, where
N is the number of turns
I is the current in Amps.
So you want to get the product of N and I as high as your physical design will allow. For instance that may be 10 turns at 100A or 2000 turns at 0.5A.
Of course the wire size needed depends on your choice of current.
And the voltage needed depends on the chosen current and on the resistance (length) of wire in the winding.
Also keep in mind that the magnetic field strength decreases with the square of the distance from the coil.
A quick search found some calculators and a discussion related to the above. They have 'Area' as a variable, which seems to be the cross section area of what you are trying to pick up/move (as long as it fits in the core of the solenoid), and a variable 'g' which is the distance from the solenoid to what you are trying to attract.
Note: My knowledge of this subject is limited, so these calculators could be completely wrong too! Hopefully others here can chime in on their relevance.
https://daycounter.com/Calculators/Magnets/Solenoid-Force-Calculator.phtml
http://onlinecalculators.brainmeasures.com/Electric/SolenoidCoil.aspx
https://www.eevblog.com/forum/beginners/solenoid-force-calculations/
For further information, above found with:
https://www.google.com/search?&q=solenoid+force+calculator
Hopes this helps!
Cheers,
Tom
p.s. I can see how that external core that
@Rive suggested would help, alot, when the target object is already inside the coil by concentrating the field there. I believe however that this would reduce/eliminate the external field of the solenoid. From your brief description, it sounds like this is not what you are after.